` DEGLI STUDI DI PADOVA UNIVERSITA Sede Amministrativa: Universit`a degli Studi di Padova

Dipartimento di Matematica Pura ed Applicata

SCUOLA DI DOTTORATO DI RICERCA IN SCIENZE MATEMATICHE INDIRIZZO DI MATEMATICA COMPUTAZIONALE CICLO XXI

ON MATRICES WITH THE EDMONDS-JOHNSON PROPERTY

Direttore della Scuola: Prof. Bruno Chiarellotto Supervisore: Dott. Giacomo Zambelli

Dottorando: Dott. Alberto Del Pia

To my parents

Acknowledgments I would like to express my sincere appreciation to my advisor Giacomo Zambelli, for his guidance, encouragement, and continuous support through the course of this work. His extensive knowledge, vision, and creative thinking have been the source of inspiration for me throughout this work. I am very grateful to Antoine Musitelli for his great alternative ideas, which have undoubtedly augmented this work. I also want to express my gratitude towards Carlo Filippi, who introduced me to the field of combinatorics when I was an undergraduate student, and encouraged me to continue. Special thanks go to Michele Conforti, for his fundamental advice and help.

Abstract The strong Chv´atal rank of a rational matrix A is the smallest number t such that the polyhedron defined by the system b ≤ Ax ≤ c, l ≤ x ≤ u has Chv´atal rank at most t for all integral vectors b, c, l, u. Matrices with strong Chv´atal rank at most 1 are said to have the Edmonds-Johnson property, since it was shown by PEdmonds and Johnson [16] that any integral matrix A = (αij ) such that i |αij | ≤ 2 for each column index j has Chv´atal rank at most 1. While the class of integral matrices with strong Chv´atal rank 0 is well understood, since it is the class of totally unimodular matrices, no general characterization is known for integral matrices with the Edmonds-Johnson property. Another class of matrices known to have the Edmonds-Johnson property is the class of the edge-node incidence matrices of bidirected graphs with no odd-K4 minors (Gerards and Schrijver [19]). The matrices in such class, as well as the ones considered by Edmonds and Johnson in [16], are totally half-modular, that is, they are integral matrices such that for any nonsingular square submatrix B, 2B −1 is integral. Gerards and Schrijver posed the question of which totally half-modular matrices have the Edmonds-Johnson property, and proposed a characterization of this class in terms of excluded minors [18]. As far as we know, this question is wide open. In Chapter 1 we provide some definitions and results that will be needed later. In Chapter 2 we introduce the Edmonds-Johnson property, and survey some related results. Our contributions are presented in the remaining two chapters. In Chapter 3, we study systems of the from b ≤ Mx ≤ c l ≤ x ≤ u,

(1)

viii

Abstract

for integral vectors b, c, l, u, where M is obtained from a totally unimodular matrix with two nonzero elements per row by multiplying by 2 some of its columns. The case where M is obtained from the transpose of the incidence matrix of a bipartite graph by multiplying by 2 some of the columns, has been studied by Conforti et al. in [7]. In this case, they derived an explicit characterization of the inequalities defining the integer hull, and showed that the problem of maximizing a linear function cx, with c integral, over the integer hull of (1) can be solved in strongly polynomial time. We give an explicit description of a totally dual integral system that describes the integer hull of the polyhedron P defined by (1). Since the inequalities of such totally dual integral system are Chv´atal inequalities for P , this implies that the matrix M has the Edmonds-Johnson property. We also derive a strongly polynomial time algorithm to find an integral optimal dual solution for the problem of maximizing a linear function with integer coefficients over the totally dual integral system describing the integer hull of (1). The results in Chapter 3 are joint work with G. Zambelli [11]. In Chapter 4 we study totally half-modular matrices obtained from matrices 0, ±1 with at most two nonzero entries per column by multiplying by 2 some of the columns. We show that the matrix   1 1 2 0 M4 =  −1 0 0 2  0 1 0 2 is the only minor minimal matrix in such class (up to multiplying rows and columns by −1) that does not have the Edmonds-Johnson property. We will give a formal definition of minor in Chapter 2. The theorem of Edmonds and Johnson discussed above is the special case where the sum of the absolute values of the entries in each column is at most 2. We will also show that, for each matrix M in this class that does not contain M4 as a minor, one can minimize in polynomial time any linear function over the integer hull of b ≤ M x ≤ c, l ≤ x ≤ u, for all integral vectors b, c, l, u. The results in Chapter 4 are joint work with A. Musitelli and G. Zambelli [10]. A partial result was shown by Del Pia and Zambelli [12].

Sommario Il rango forte di Chv´atal di una matrice razionale A `e il pi` u piccolo numero t tale che il poliedro definito dal sistema b ≤ Ax ≤ c, l ≤ x ≤ u ha rango di Chv´atal al pi` u t per tutti i vettori interi b, c, l, u. Matrici con rango forte di Chv´atal al pi` u 1 si dicono avere la propriet` a di Edmonds-Johnson, poich´e fu mostrato da Edmonds e Johnson [16] che ogni matrice intera A = (αij ) tale P che i |αij | ≤ 2 per ogni indice di colonna j ha rango di Chv´atal al pi` u 1. Mentre la classe di matrici intere con rango forte di Chv´atal 0 `e ben caratterizzata, poich´e `e la classe delle matrici totalmente unimodulari, non `e nota una caratterizzazione generale per matrici intere con la propriet`a di Edmonds-Johnson. Un’altra classe nota di matrici con la propriet`a di Edmonds-Johnson `e la classe delle matrici di incidenza arco-nodo di grafi biorientati senza minori odd-K4 (Gerards e Schrijver [19]). Le matrici in questa classe, come quelle considerate da Edmonds e Johnson in [16], sono totalmente 12 -modulari, ovvero sono matrici intere tali che per ogni sottomatrice quadrata non singolare B, 2B −1 `e intera. Gerards e Schrijver posero la domanda di quali matrici totalmente 12 -modulari hanno la propriet`a di Edmonds-Johnson, e proposero una caratterizzazione di questa classe in termini di minori esclusi [18]. Per quanto ne sappiamo, questa domanda `e ancora aperta. Nel Capitolo 1 forniamo alcune definizioni e risultati che saranno necessarie pi` u avanti. Nel Capitolo 2 introduciamo la propriet`a di Edmonds-Johnson, ed esaminiamo qualche risultato correlato. I nostri contributi sono presentati nei rimanenti due capitoli. Nel Capitolo 3, studiamo sistemi nella forma b ≤ Mx ≤ c l ≤ x ≤ u,

(2)

x

Sommario

per vettori interi b, c, l, u, dove M `e ottenuta da una matrice totalmente unimodulare con due elementi diversi da zero per riga moltiplicando per 2 alcune colonne. Il caso in cui M `e ottenuta dalla trasposta della matrice di incidenza di un grafo bipartito moltiplicando per 2 alcune colonne, `e stato studiato da Conforti et al. in [7]. In questo caso, loro hanno derivato una caratterizzazione esplicita delle disuguaglianze che definiscono l’inviluppo convesso dei punti interi, e hanno mostrato che il problema di massimizzare una funzione lineare cx, con c intero, nell’inviluppo convesso dei punti interi di (2) pu`o essere risolto in tempo fortemente polinomiale. Noi diamo una descrizione esplicita di un sistema totally dual integral che descrive l’inviluppo convesso dei punti interi del poliedro P definito da (2). Dato che le disuguaglianze di tale sistema totally dual integral sono disuguaglianze di Chv´atal per P , questo implica che la matrice M ha la propriet`a di Edmonds-Johnson. Inoltre deriviamo un algoritmo fortemente polinomiale per trovare una soluzione duale ottima intera per il problema di massimizzare una funzione lineare con coefficienti interi nel sistema totally dual integral che descrive l’inviluppo convesso dei punti interi di (2). I risultati nel Capitolo 3 sono ottenuti in collaborazione con G. Zambelli [11]. Nel Capitolo 4 studiamo matrici totalmente trici 0, ±1 con al pi` u due elementi non zero per alcune colonne. Mostriamo che la matrice  1 1 2 0  −1 0 0 2 M4 = 0 1 0 2

1 -modulari 2

ottenute da macolonna moltiplicando per 2  

`e l’unica matrice minimale rispetto alla relazione minore in tale classe (a meno di moltiplicare righe e colonne per −1) che non ha la propriet`a di Edmonds-Johnson. Daremo una definizione formale di minore nel Capitolo 2. Il teorema di Edmonds e Johnson discusso sopra `e il caso particolare in cui la somma dei valori assoluti degli elementi in ogni colonna `e al pi` u 2. Mostreremo anche che, per ogni matrice M in questa classe che non contiene M4 come minore, si pu`o minimizzare in tempo polinomiale ogni funzione lineare nell’inviluppo convesso dei punti interi di b ≤ M x ≤ c, l ≤ x ≤ u, per tutti i vettori interi b, c, l, u. I risultati nel Capitolo 4 sono ottenuti in collaborazione con A. Musitelli e G. Zambelli [10]. Un risultato parziale `e stato mostrato da Del Pia e Zambelli [12].

Contents Abstract

vii

Sommario 1 Introduction 1.1 Graph theory . . . . . . . . . . . . 1.1.1 Undirected graphs . . . . . 1.1.2 Directed graphs . . . . . . . 1.1.3 Bidirected graphs . . . . . . 1.2 Polyhedra and linear inequalities . 1.2.1 Polyhedra and polytopes . . 1.2.2 Faces . . . . . . . . . . . . . 1.2.3 Minimal faces and vertices . 1.2.4 A polynomial result in linear 1.3 Integral polyhedra . . . . . . . . . 1.3.1 Totally unimodular matrices 1.3.2 Total dual integrality . . . . 1.3.3 Chv´atal-Gomory cuts . . . .

ix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 The Edmonds-Johnson property 2.1 Classes of matrices with the Edmonds-Johnson 2.1.1 The Edmonds and Johnson’s class . . . 2.1.2 The Gerards and Schrijver’s class . . . 2.2 A conjecture by Gerards and Schrijver . . . .

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property . . . . . . . . . . . . . . . . . .

3 Bipartite vertex covers with parity conditions 3.1 Introduction . . . . . . . . . . . . . . . . . . . . 3.2 Bipartite case . . . . . . . . . . . . . . . . . . . 3.2.1 The extended graph . . . . . . . . . . . 3.2.2 Proof of Theorem 3.4 . . . . . . . . . . . 3.3 General case . . . . . . . . . . . . . . . . . . . .

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1 . 1 . 1 . 3 . 4 . 5 . 5 . 5 . 6 . 6 . 7 . 7 . 10 . 12

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15 17 17 23 26

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31 31 34 35 36 42

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xii

CONTENTS 3.4

Polynomial time solvability . . . . . . . . . . . . . . . . . . . . 44 3.4.1 Bipartite case . . . . . . . . . . . . . . . . . . . . . . . 44 3.4.2 General case . . . . . . . . . . . . . . . . . . . . . . . . 47

4 A class of matrices arising from bidirected graphs 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 4.2 First Chv´atal closure . . . . . . . . . . . . . . . . . 4.2.1 Algorithmic aspects . . . . . . . . . . . . . . 4.3 Balanced bipartitions . . . . . . . . . . . . . . . . . 4.4 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . 4.4.1 Half-integrality for some special cases . . . . 4.4.2 Structure of (G, F ) . . . . . . . . . . . . . . 4.4.3 x¯ is half-integral . . . . . . . . . . . . . . . . 4.4.4 Shrinkable pairs of edges . . . . . . . . . . . 4.4.5 The end: finding a balanced bipartition . . .

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49 49 53 58 59 65 70 74 85 89 94

Chapter 1 Introduction 1.1

Graph theory

In this thesis we will work with graphs. Therefore, in this section we give a brief review of some elementary concepts in graph theory. We refer the reader to West [34] for graph theory notations and for proofs of the mentioned results.

1.1.1

Undirected graphs

An (undirected) graph is a pair G = (V, E), where V is a finite set, and E is a multiset of unordered pairs of elements of V . The elements of V are called the nodes of G, and the elements of E are called the edges of G. We denote with vw an edge {v, w} in E. The nodes v, w are called the endnodes of vw. If v = w then the edge vw = vv is called a loop. Given a graph G, we denote with V (G) the set of nodes of G, with E(G) the set of edges of G, and with L(G) the set of loops in E(G). The term ‘multiset’ in the definition of graph means that a pair of nodes may occur several times in E. A pair occurring more than once in E is called a parallel edge. So distinct edges may be represented in E by the same pair. Nevertheless, we shall often speak of ‘an edge vw’ or even of ‘the edge vw’, where ‘an edge with endnodes v, w’ would be more correct. A loopless graph is a graph having no loops, a simple graph is a graph having no loops or parallel edges. We shall say that an edge vw connects the nodes v and w. The nodes v and w are adjacent if there is an edge connecting v and w. The edge vw is said to be incident with the node v and with the node w, and conversely. Two edges e and f are adjacent if they have an endnode in common. We say that a node is isolated if it is not incident with any edge.

2

Introduction

If U ⊆ V , then δG (U ) (or δ(U ) when there is no ambiguity) denotes the set of edges in E that have exactly one endnode in U . Notice that e ∈ δG (U ) for every loop e with endnode in U . When v is a node of G, we will write δ(v) instead of δ({v}). The number |δG (v)| is the degree of v in G. A graph G0 = (V 0 , E 0 ) is a subgraph of G = (V, E) if V 0 ⊆ V and E 0 ⊆ E. If E 0 is the multiset of all edges of G with all endnodes in V , then G0 is said to be induced by V 0 , and we denote G0 by G[V 0 ], or by G \ V 00 , where V 00 = V \ V 0 . When v is a node of G, we will write G \ v instead of G \ {v}. If V = V 0 , then we denote G0 by G \ E 00 , where E 00 = E \ E 0 . When e is an edge of G, we will write G \ e instead of G \ {e}. A walk in the graph G = (V, E) from v0 to vt , is a sequence of the form W = v0 , e1 , v1 , e2 , . . . , vt−1 , et , vt

(1.1)

where v0 , . . . , vt are nodes and e1 , . . . , et are edges, such that ei = vi−1 vi for i = 1, . . . , t. If there is no ambiguity, we often write walk W just as a sequence of nodes, i.e. W = v0 , v1 , . . . , vt . The nodes v0 and vt are the endnodes of the walk, and we say that the walk starts in v0 and ends in vt . We call walk W a v0 − vt -walk, and it is said to connect v0 and vt . We define V (W ) = {v0 , . . . , vt }, and E(W ) = {e1 , . . . , et }. Walk W is a trail if all edges in (1.1) are different. A trail T 0 is a subtrail of a trail T if T 0 is a subsequence of T , say T 0 = vi , ei+1 , vi+1 , ei+2 , . . . , vj−1 , ej , vj , i < j, such that V (T 0 ) = {vh : i ≤ h ≤ j}, E(T 0 ) = {eh : i + 1 ≤ h ≤ j}. Walk W is a path if all nodes and edges in (1.1) are different. If P is a path, a subtrail of P is called a subpath of P . The length of walk W is t. The distance between two nodes r and s in a graph is the minimum length of a path connecting r and s. If v0 = vt , walk W is called closed. A closed walk of length at least two and without repeated edges or nodes (except for the endnodes) is called a cycle. So, for the purpose of this thesis, loops are not cycles. An edge connecting two nodes of a cycle which are not connected by an edge of the cycle is called a chord of the cycle. An Eulerian circuit in a graph G = (V, E) is a closed trail C such that E(C) = E. It is well known that a loopless graph G has an Eulerian circuit if and only if |δ(v)| is even for every v ∈ V . A graph is connected if each two nodes of the graph are connected by a path. The (connected) components of a graph are its maximally connected subgraphs. A (node-)cutset of a graph G is a set V 0 ⊆ V such that G \ V 0 has strictly more connected components than G. If {v} is a node-cutset of G, we call v a cutnode of G. An edge-cutset of a graph G is a set E 0 ⊆ V such that G \ E 0 has strictly more connected components than G. If {e} is an edge-cutset of G, we call e a cutedge of G. A block of a graph G is a maximal subgraph of G that does not have a cutnode.

1.1 Graph theory

3

A graph having no cycles is said to be acyclic, and is called a forest. A tree is a connected forest. It is not difficult to see that the following are equivalent for a given simple graph G = (V, E): (i) G is a tree; (ii) G contains no cycles and |E| = |V | − 1; (iii) G is connected and |E| = |V | − 1; (iv) any two nodes of G are connected by exactly one simple path. If we add one new edge connecting two nodes of the tree, we obtain a graph with a unique cycle. Each tree with at least two nodes has at least two nodes of degree one. A subgraph G0 = (V 0 , E 0 ) of G = (V, E) is a spanning (sub)tree of G if 0 V = V and G0 is a tree. Then G has a spanning tree if and only if G is connected. In a graph G = (V, E), we say that a set E 0 ⊆ E is a star centered at node v if all the edges in E 0 are incident with v ∈ V . A matching is a set of pairwise disjoint edges. A matching covering all nodes is called a perfect matching. A graph G = (V, E) is called bipartite if V can be partitioned into two classes V1 and V2 such that each edge of G contains a node in V1 and a node in V2 . The sets V1 and V2 are called sides. It is easy to see that a graph G is bipartite if and only if G contains no cycles of odd length. The incidence matrix of G is the matrix with rows and columns indexed by V and E, respectively, where the entry in position (v, e) is 1 if edge e is not a loop and it is incident with node v, is 2 if edge e is a loop incident with node v, and is 0 otherwise. Notice that the incidence matrix of a loopless graph is a {0, 1}-matrix. The edge-node incidence matrix of G is the transpose of the incidence matrix of G.

1.1.2

Directed graphs

A directed graph, or digraph, is a pair D = (V, A), where V is a finite set, and A is a finite multiset of ordered pairs of distinct elements of V . The elements of V are called the nodes, and the elements of A are called the arcs of D. The nodes v and w are called the tail and the head of the arc (v, w), respectively. So the difference with undirected graphs is that orientations are given to the pairs. Each directed graph gives rise to an underlying undirected graph,

4

Introduction

in which we forget the orientation of the arcs. When there is no ambiguity, we use ‘undirected’ terminology for directed graphs. We say that the arc (v, w) enters w and leaves v. If U is a set of nodes such that v ∈ / U and w ∈ U , then (v, w) is said to enter U and to leave V \ U . − If U ⊆ V , then δD (U ) (or δ − (U ) when there is no ambiguity) denotes the set + of arcs of D entering U , and δD (U ) (or δ + (U )) denotes the set of arcs of D − + leaving U . δ (v) and δ (v) stand for δ − ({v}) and δ + ({v}). The in-degree of v is |δ − (v)|, while the out-degree of v is |δ + (v)|. A directed walk, from v0 to vt , or a v0 − vt -walk, in a digraph D = (V, A) is a sequence of the form W = v0 , a1 , v1 , a2 , . . . , vt−1 , at , vt

(1.2)

where v0 , . . . , vt are nodes and a1 , . . . , at are arcs, such that ai = (vi−1 , vi ) for i = 1, . . . , t. If there is no ambiguity, we often write directed walk W just as a sequence of nodes, i.e. W = v0 , v1 , . . . , vt . Walk W is said to start in v0 and to end in vt . The nodes v0 and vt are the endnodes of the walk. The number t is the length of the walk W . Directed walk W is a directed trail if all arcs in (1.2) are different. Directed walk W is a directed path if all nodes and arcs in (1.2) are different. A v0 − vt -walk is closed if v0 = vt . A directed cycle is a closed directed walk of length at least two, without repeated nodes or arcs (except for its endnodes). We call acyclic a digraph with no directed cycles. An (undirected) walk (resp. trail, path, cycle) is a walk (resp. trail, path, cycle) in the underlying undirected graph. In a natural way, an undirected walk or cycle in a directed graph has forward arcs and backward arcs. A digraph D is connected if its underlying undirected graph is connected. The incidence matrix of a digraph D = (V, A) is the matrix with rows and columns indexed by V and A, respectively, where the entry in position (v, a) is +1 if v is the head of a, −1 if v is the tail of a, or 0 otherwise. The edge-node incidence matrix of D is the transpose of the incidence matrix of D.

1.1.3

Bidirected graphs

A bidirected graph is a triple G = (V, E, σ), where (V, E) is an undirected graph (possibly with loops and parallel edges) and σ is a signing of (V, E), i.e. a map that assigns to each e ∈ E and v ∈ e a sign σv,e ∈ {+1, −1}. We say that a nonloop edge e = vw of G is a ++ edge if σv,e = σw,e = +1, it is a −− edge if σv,e = σw,e = −1, and it is a +− edge otherwise. For convenience, we define σv,e := 0 if v ∈ / e. The odd edges of G are the ++ edges and −−

1.2 Polyhedra and linear inequalities

5

edges. Each bidirected graph gives rise to an underlying undirected graph, in which we forget σ. Sometimes, when no misunderstanding is possible, we use ‘undirected’ terminology for bidirected graphs. A cycle C in G, is even if the number of odd edges in it is even. This is equivalent P to saying that the sum of the signs on the edges in C is divisible by 4 ( vw∈C (σv,vw + σw,vw ) ≡4 0). Otherwise we say that C is odd. We call a bidirected graph bipartite if it does not contain any odd cycle. G is connected if its underlying undirected graph is connected. The sign matrix of a bidirected graph G is the V × E matrix Σ(G) = (σv,e ). The incidence matrix of G is obtained from the sign matrix of G by multiplying by 2 the columns corresponding to the loops of G. The edge-node incidence matrix of G is the transpose of the incidence matrix of G.

1.2

Polyhedra and linear inequalities

In this section we introduce the fundamental notion of polyhedra. We refer the reader to Schrijver [29] for a more detailed introduction and for the proofs of the following theorems.

1.2.1

Polyhedra and polytopes

A set P of vectors in Rn is called a (convex) polyhedron if P = {x : Ax ≤ b}

(1.3)

for some matrix A and vector b. Thus P is a polyhedron if it is the intersection of finitely many affine half-spaces, where an affine half-space is a set of the form {x : ax ≤ β} for some nonzero row vector a and some number β. If (1.3) holds, we say that Ax ≤ b defines or determines P . The convex hull of a set X of vectors is the inclusionwise minimal convex set containing X, and is denoted by conv(X). A set of vectors is a (convex) polytope if it is the convex hull of finitely many vectors. The following theorem gives a fundamental link between polytopes and polyhedrons. Theorem 1.1. (Minkowski-Weyl theorem for polytopes.) A set P is a polytope if and only if P is a bounded polyhedron.

1.2.2

Faces

If c is a nonzero vector, P is a polyhedron, and δ = max{cx : x ∈ P }, the affine hyperplane {x : cx = δ} is called a supporting hyperplane of P . A

6

Introduction

subset F of P is called a face of P if F = P or if F is the intersection of P with a supporting hyperplane of P . Theorem 1.2. F is a face of P if and only if F is nonempty and F = {x ∈ P : A0 x = b0 } for some subsystem A0 x ≤ b0 of Ax ≤ b. It is known that the dimension of F is n − rk(A0 ), where rk(A0 ) denotes the rank of A0 . It follows that: (i) P has only finitely many faces; (ii) each face is a nonempty polyhedron; (iii) if F is a face of P and F 0 ⊆ F , then: F 0 is a face of P if and only if F 0 is a face of F .

1.2.3

Minimal faces and vertices

A minimal face of P is a face not containing any other face. It is well known that a face F of P is a minimal face if and only if F is an affine subspace. In fact, there is the following result of Hoffman and Kruskal [25]. Theorem 1.3. A set F is a minimal face of P , if and only if ∅ 6= F ⊆ P and F = {x : A0 x = b0 } for some subsystem A0 x ≤ b0 of Ax ≤ b. In particular, all minimal faces of P have the same dimension, namely n minus the rank of A. If each minimal face of P consists of just one point, then P is called pointed. These points (or these minimal faces) are called the vertices of P . So each vertex is determined by n linearly independent equations from the systems Ax = b. Remark 1.4. A vector x¯ ∈ P is a vertex of P if and only if it satisfies n linearly independent equations of the system Ax = b. Moreover, a vector x ∈ P is a vertex of P if and only if x cannot be expressed as a convex combination of vectors in P .

1.2.4

A polynomial result in linear programming

An interesting result was obtained by Tardos [32, 33], who showed that any linear program max{cx : Ax ≤ b} can be solved in at most p(size(A)) elementary arithmetic operations on numbers of size polynomially bounded by size(A, b, c), where p is a polynomial. Thus the sizes of b and c do not contribute in the number of arithmetic steps. In particular the class of LPproblems where A is a {0, ±1}-matrix has a strongly polynomial algorithm.

1.3 Integral polyhedra

7

Theorem 1.5. There exists an algorithm which solves a given rational LPproblem max{cx : Ax ≤ b} in at most P (size(A)) elementary arithmetic operations on numbers of size polynomially bounded by size(A, b, c), for some polynomial P . Corollary 1.6. There exists a strongly polynomial algorithm for rational LP-problems with {0, ±1}-constraint matrix.

1.3

Integral polyhedra

A vector or matrix is called integral if its entries are all integers. Define, for any polyhedron P , the integer hull PI of P by PI := conv(P ∩ Zn ), i.e. the convex hull of the integral vectors in P . So, given a rational matrix A, and rational vectors b and c, determining max{cx : Ax ≤ b, x integral} is equivalent to determining max{cx : x ∈ PI }, for P := {x : Ax ≤ b}. We say that a rational polyhedron is integral if each face contains integral vectors. This is equivalent to saying that the polyhedron is the convex hull of the integral vectors contained in it. We state the following result of Meyer [28], which is trivial if P is bounded. Theorem 1.7. For any rational polyhedron P , the set PI is also a polyhedron.

1.3.1

Totally unimodular matrices

An integral matrix is unimodular if it has determinant ±1. In this section we consider totally unimodular matrices, which are matrices with all subdeterminants equal to +1, −1, or 0 (in particular, each entry is +1, −1, or 0). Notice that A is totally unimodular if and only if the transpose of A is totally unimodular. Totally unimodular matrices yield a prime class of linear programming problems with integral optimum solutions. In the present section we introduce the basic theory and examples of totally unimodular matrices. We state Hoffman and Kruskal’s characterization of totally unimodular matrices, providing the link between total unimodularity and integer linear programming. Then we give some further characterizations, and we discuss some classes of examples of totally unimodular matrices. Theorem 1.8. (Hoffman and Kruskal’s theorem). Let A be an integral matrix. Then A is totally unimodular if and only if for each integral vector b the polyhedron {x : x ≥ 0, Ax ≤ b} is integral.

8

Introduction Since for any totally unimodular matrix A also the matrix   I  −I     A  −A

is totally unimodular, it follows that an integral matrix A is totally unimodular if and only if for all integral vectors b, c, l, u the vertices of the polytope {x : l ≤ x ≤ u, b ≤ Ax ≤ c} are integral. Similarly, one may derive from Theorem 1.8 that an integral matrix A is totally unimodular if and only if one of the following polyhedra is integral, for each integral vector b and for some integral vector c: {x : x ≤ c, Ax ≤ b}, {x : x ≤ c, Ax ≥ b}, {x : x ≥ c, Ax ≤ b}, {x : x ≥ c, Ax ≥ b}. Formulated in terms of linear programming, Theorem 1.8 says the following. Corollary 1.9. An integral matrix A is totally unimodular if and only if for all integral vectors b and c both sides of the linear programming duality equation max{cx : x ≥ 0, Ax ≤ b} = min{yb : y ≥ 0, yA ≥ c} are achieved by integral vectors x and y (if they are finite). There are other characterizations of totally unimodular matrices. We collect some of them, together with the above characterization, in the following theorem. Characterizations (ii) and (iii) are due to Hoffman and Kruskal [25], characterization (iv) to Ghouila-Houri [20], characterization (v) to Camion [1, 2, 3]. Theorem 1.10. Let A be a matrix with entries 0, +1, or −1. Then the following are equivalent: (i) A is totally unimodular, i.e. each square submatrix of A has determinant 0, +1, or −1; (ii) for each integral vector b the polyhedron {x : x ≥ 0, Ax ≤ b} has only integral vertices; (iii) for all integral vectors b, c, l, u the polyhedron {x : l ≤ x ≤ u, b ≤ Ax ≤ c} has only integral vertices;

1.3 Integral polyhedra

9

(iv) each collection of columns of A can be partitioned into two sets so that the sum of the columns in one set minus the sum of the columns in the other set is a vector with entries 0, +1, and −1; (v) the sum of the entries in any square submatrix with even row and column sums is divisible by four. We give three well known examples of totally unimodular matrices due to Hoffman and Kruskal [25], and Heller and Tompkins [24]. Example 1. Bipartite graphs Let G = (V, E) be an undirected graph, and let M be the V × E-incidence matrix of G. Then: Theorem 1.11. M is totally unimodular if and only if G is bipartite. So M is totally unimodular if and only if the rows of M can be split into two classes so that each column contains a 1 in each of these classes. Theorem 1.11 easily follows from Ghouila-Houri’s characterization of totally unimodular matrices ((iv) in Theorem 1.10). Example 2. Directed graphs Let D = (V, A) be a directed graph, and let M be the V ×A-incidence matrix of D. Then M is totally unimodular. In other words: Theorem 1.12. A {0, ±1}-matrix with in each column exactly one +1 and exactly one −1 is totally unimodular. Theorem 1.12 follows directly from Ghouila-Houri’s characterization ((iv) in Theorem 1.10). Example 3. A combination of Examples 1 and 2 Theorem 1.13. Let M be a {0, ±1}-matrix with exactly two nonzeros in each column. Let G be the bidirected graph whose incidence matrix is M . Then the following are equivalent: (i) M is totally unimodular; (ii) the rows of M can be split into two classes such that for each column if the two nonzeros in the column have the same sign then they are in different classes, and if they have opposite sign then they are both in one and the same class;

10

Introduction

(iii) G is bipartite. This follows easily from Example 2 (multiply all rows in one of the classes by −1, and we obtain a {0, ±1}-matrix with in each column exactly one +1 and exactly one −1) and from (iv) and (v) of Theorem 1.10. Conversely, it includes Examples 1 and 2.

1.3.2

Total dual integrality

In Section 1.3.1 we introduced totally unimodular matrices, which are exactly those integral matrices A with the property that the polyhedron P := {x ≥ 0 : Ax ≤ b} is integral for each integral vector b. In this section we fix A and b, and we study integral polyhedra and the related notion of total dual integrality. We refer the reader to [29] for an extensive treatment on total dual integrality. The basis for this section is the following result of Edmonds and Giles [14]. Theorem 1.14. A rational polyhedron P is integral, if and only if each rational supporting hyperplane of P contains integral vectors. The following is an equivalent formulation. Corollary 1.15. Let Ax ≤ b be a system of rational linear inequalities. Then max{cx : Ax ≤ b} is achieved by an integral vector x for each vector c for which the maximum is finite, if and only if max{cx : Ax ≤ b} is an integer for each integral vector c for which the maximum is finite. We call a rational system Ax ≤ b of linear inequalities totally dual integral, abbreviated TDI, if the minimum in the LP-duality equation max{cx : Ax ≤ b} = min{yb : y ≥ 0, yA = c} has an integral optimum solution y for each integral vector c for which the minimum is finite. So by Corollary 1.9, if A is totally unimodular, then Ax ≤ b is TDI for each rational vector b. Edmonds and Giles [14] showed that total dual integrality of Ax ≤ b implies that also the maximum has an integral optimum solution, if b is integral. Corollary 1.16. If Ax ≤ b is a TDI-system and b is integral, the polyhedron {x : Ax ≤ b} is integral.

1.3 Integral polyhedra

11

Note however that total dual integrality is not a property of just polyhedra. The systems     µ ¶ µ ¶µ ¶ µ ¶ 1 1 0 x 1 1 x 0 1 1  1 −1  ≤  0  and ≤ x2 1 −1 x2 0 1 0 0 define the same polyhedron, but the first system is TDI and the latter not. Generally a TDI-system contains more constraints than necessary for just defining the polyhedron. The following Theorem basically states that each face of a TDI-system is TDI. Theorem 1.17. Let Ax ≤ b, ax ≤ β be a TDI-system. Then the system Ax ≤ b, ax = β is also TDI. We give a proof of this result from Schrijver [29]. Proof. Let c be an integral vector, with max{cx : Ax ≤ b, ax = β} = min{yb + (λ − µ)β : y ≥ 0, λ, µ ≥ 0, yA + (λ − µ)a = c}

(1.4)

finite. Let x∗ , y ∗ , λ∗ , µ∗ attain these optima (possibly being fractional). Let c0 := c + N a where N is an integer satisfying N ≥ µ∗ − λ∗ and N a integer. Then the optima max{c0 x : Ax ≤ b, ax ≤ β} = min{yb + λβ : y ≥ 0, λ ≥ 0, yA + λa = c0 }

(1.5)

are finite, since x := x∗ is a feasible solution for the maximum, and y := y ∗ , λ := λ∗ + N − µ∗ is a feasible solution for the minimum. Since Ax ≤ b, ax ≤ β is TDI, the minimum (1.5) has an integer optimum ˜ Then y := y˜, λ := λ, ˜ µ := N is an integer optimum solution, say, y˜, λ. solution for the minimum in (1.4): it is feasible in (1.4), and it is optimum as: ˜ − N )β = y˜b + λβ ˜ − Nβ y˜b + (λ ≤ y ∗ b + (λ∗ + N − µ∗ )β − N β = y ∗ b + (λ∗ − µ∗ )β. (Here ≤ follows from the fact that y ∗ , λ∗ + N − µ∗ is a feasible solution for ˜ is an optimum solution for this minimum.) the minimum in (1.5), and y˜, λ So the minimum in (1.4) has an integral optimum solution. The following is an important result of Giles and Pulleyblank [21].

12

Introduction

Theorem 1.18. For each rational polyhedron P there exists a TDI-system Ax ≤ b with A integral and P = {x : Ax ≤ b}. Here b can be chosen to be integral if and only if P is integral. Chandrasekaran [4] showed the following result. Theorem 1.19. Let Ax ≤ b be a TDI-system with A integral, and let c be an integral vector. Then an integral optimum solution for min{yb : y ≥ 0, yA = c} can be found in polynomial time.

1.3.3

Chv´ atal-Gomory cuts

Let H be a rational affine half-space {x : cx ≤ δ}, where c is a nonzero vector whose components are relatively prime integers (each rational affine half-space can be represented in this way), then clearly HI = {x : cx ≤ bδc}. Geometrically, HI arises from H by shifting the bounding hyperplane of H until it contains integral vectors. Now define for any polyhedron P : P 0 := ∩H⊇P HI where the intersection ranges over all rational affine half-spaces H with H ⊇ P . (Clearly, we may restrict the intersection to half-spaces whose bounding hyperplane is a supporting hyperplane of P.) The polyhedron P 0 is called the first (Chv´atal) closure of P . As P ⊆ H implies PI ⊆ HI , it follows that PI ⊆ P 0 . So P ⊇ P 0 ⊇ P 00 ⊇ · · · ⊇ PI . The half-spaces HI are called Chv´ atal-Gomory cuts, while their defining inequalities are called Chv´ atal(-Gomory) inequalities. A constraint is valid for a set S if each element in S satisfies this constraint. Algebraically, if P = {x : Ax ≤ b}, then any rational valid inequality for P is of the form (λA)x ≤ λb with λ ≥ 0, where we can assume that λA is integral. Therefore any Chv´atal-Gomory inequality can be written in the form (λA)x ≤ bλbc, where λ ≥ 0 is such that λA is integral. The vector λ is called a Chv´atal-Gomory multiplier. The first (Chv´atal) closure of a system Ax ≤ b, is the first closure of the polyhedron defined by such system. We say that a Chv´atal inequality for a system of linear inequalities is nontrivial if it is not implied by such system. Two inequalities αx ≤ β and α0 x ≤ β 0 valid for the first Chv´atal closure of a system of linear inequalities are equivalent if they define the same face of the first Chv´atal closure. Lemma 1.20. If A and b are integral, any nontrivial Chv´atal inequality for Ax ≤ b is equivalent to an inequality of the form (λA)x ≤ bλbc such that 0 ≤ λ < 1, λA is integral, and λb is not integral.

1.3 Integral polyhedra

13

Proof. By definition, any Chv´atal-Gomory inequality for P can be written in the form (λA)x ≤ bλbc, where λ ≥ 0 is such that λA is integral. Since (λ − bλc)A is integral, then (λ − bλc)Ax ≤ b(λ − bλc)bc is valid for the first Chv´atal closure of Ax ≤ b. Furthermore (λA)x ≤ bλbc is the sum of (λ − bλc)Ax ≤ b(λ − bλc)bc and bλcAx ≤ bλcb, where the last inequality is implied by Ax ≤ b. Clearly, if λb is integral, then the Chv´atal inequality (λA)x ≤ bλbc is implied by the system Ax ≤ b. Theorem 1.21. For any rational polyhedron P , P 0 is a polyhedron again. In general, it was shown by Eisenbrand [17] that optimizing over the first Ch´atal closure of a system of linear inequalities is N P-complete. Theorem 1.22. For each rational polyhedron P there exists a number t such that P (t) = PI . 0

(Here: P (0) := P, P (t+1) := P (t) .) A direct consequence of Theorem 1.22 is Theorem 1.14: if each rational supporting hyperplane of a rational polyhedron P contains integral vectors, then P = P 0 and hence P = PI . This theorem also implies a result of Chv´atal [5] for not-necessarily rational polytopes: Corollary 1.23. For each polytope P there exists a number t such that P (t) = PI . The Chv´atal rank of a polyhedron P is the smallest number t such that P = PI . The Chv´atal rank of a system of linear inequalities is the Chv´atal rank of the polyhedron defined by such system. The following Lemma will be useful in the next Chapters. (t)

Lemma 1.24. Consider the systems

and

ax ≤ β Ax ≤ b

(1.6)

ax + s = β Ax ≤ b s ≥ 0.

(1.7)

Where A is an integral matrix, b and a are integral vectors, and β is integer. Then: x, x¯) is in the first (i) x¯ is in the first closure of (1.6) if and only if (β − a¯ closure of (1.7).

14

Introduction

(ii) (1.6) has Chv´atal rank at most 1 if and only if (1.7) has Chv´atal rank at most 1. Proof. (i): Let x¯ be in the first closure of (1.6). By contradiction assume that (β − a¯ x, x¯) is not in the first closure of (1.7). Since (β − a¯ x, x¯) satisfies (1.7), there exists Chv´atal inequality for (1.7) not satisfied by (β − a¯ x, x¯). Such inequality can be written in the form (λa+µA)x+(λ−γ)s ≤ bλβ +µbc, where λa + µA and λ − γ are integral. By substituting s = β − ax, and since λ − γ is integer, it follows that x¯ does not satisfy (µA + γa)x ≤ bµb + γβc, which is a Chv´atal inequality for (1.6), a contradiction. The opposite implication is trivial, as each Chv´atal inequality for (1.6) is also a Chv´atal inequality for (1.7). (ii): Assume that (1.6) has Chv´atal rank at most 1, and let (¯ s, x¯) be a vector in the first closure of (1.7). P Notice that s¯ = β − a¯ x. By (i), x¯ is in the first closure of (1.6), thus x¯ = i∈I yi , where yi is integral and satisfies (1.6) for every i ∈ I. Notice that P (β − ayi , yi ) is integral and satisfies (1.7) for every i ∈ I. Since (¯ s, x¯) = i∈I (β − ayi , yi ), then (1.7) has Chv´atal rank at most 1. Conversely, assume that (1.7) has Chv´atal rank at most 1, and let x¯ be a vector in the first closureP of (1.6). By (i), (β − a¯ x, x¯) is in the first closure of (1.7), thus (β − a¯ x, x¯) = i∈I (zi , yiP ), where zi and yi are integral and satisfy (1.7) for every i ∈ I. Hence x¯ = i∈I yi , where yi is integral and satisfies (1.6) for every i ∈ I. Hence (1.6) has Chv´atal rank at most 1. We state a result of Cook, Gerards, Schrijver, and Tardos [8]. Theorem 1.25. For each rational matrix A there exists a number t such that for each column vector b one has: {x : Ax ≤ b}(t) = {x : Ax ≤ b}I .

Chapter 2 The Edmonds-Johnson property Theorem 1.25 states that for each rational matrix A there exists a number t such that for each column vector b one has: {x : Ax ≤ b}(t) = {x : Ax ≤ b}I . This motivates the following definition. The strong Chv´atal rank (or cutrank ) of a rational matrix A is the smallest number t such that the polyhedron defined by the system b ≤ Ax ≤ c, l ≤ x ≤ u has Chv´atal rank at most t for all integral vectors b, c, l, u. So Theorem 1.25 states that the (strong) Chv´atal rank is a well-defined integer. It follows from Theorem 1.8 that an integral matrix A has strong Chv´atal rank 0 if and only if A is totally unimodular. Similar characterizations for higher Chv´atal ranks are not known. Matrices with strong Chv´atal rank at most 1 are said to have the Edmonds-Johnson property. In the following theorem we give some operations that preserve the Edmonds-Johnson property. Theorem 2.1. The class of matrices with the Edmonds-Johnson property is closed under the following operations: (i) permuting rows and columns; (ii) multiplying rows and columns by −1; (iii) deleting rows and columns; (iv) dividing by k ∈ N, k ≥ 2 a row where all entries are multiple of k;

16

The Edmonds-Johnson property µ

1 g f D

¶

(v) pivoting on a 1 entry, i.e. replacing matrix by the matrix µ ¶ −1 g , where f is a column vector and g a row vector. f D − fg Proof. Let A be a matrix with the Edmonds-Johnson property. By definition, the system b ≤ Ax ≤ c, l ≤ x ≤ u has Chv´atal rank at most 1 for all integral vectors b, c, l, u. (i),(ii): If A0 is obtained from A by permuting rows or columns, or multiplying them by −1, then trivially A0 has the Edmonds-Johnson property. (iii): If A0 is obtained from A by deleting a column, say corresponding to variable xj , by taking lj = uj = 0, it follows trivially that A0 has the Edmonds-Johnson property. If A0 is obtained from A by deleting a row, say the i-th row, by taking bi = −∞, ci = +∞, it follows trivially that A0 has the Edmonds-Johnson property. (iv): Assume that A0 is obtained from A by dividing by k a row, say the ith, where all entries are multiple of k. The system b0 ≤ A0 x ≤ c0 , l0 ≤ x ≤ u0 has Chv´atal rank at most 1 for all integral b0 , c0 , l0 , u0 because the system b ≤ Ax ≤ c, l0 ≤ x ≤ u0 has Chv´atal rank at most 1, where b and c are obtained from b0 and c0 by multiplying the¶i-th component by k. µ 1 g has the Edmonds-Johnson prop(v): Assume that the matrix f D erty. We want to show that the system µ ¶ µ ¶µ ¶ µ ¶ b0 −1 g x0 c0 ≤ ≤ b f D − fg x c (2.1) ¶ ¶ µ ¶ µ µ u0 x0 l0 ≤ ≤ u l x has Chv´atal rank at most 1 for all integral b0 , b, c0 , c, l0 , l, u0 , u. Let x00 = −x0 + bx. By substituting x0 = −x00 + bx in (2.1) we get ¶ µ ¶ µ ¶µ 0 ¶ µ l0 1 g x0 u0 ≤ ≤ c b f D x (2.2) ¶ ¶ µ 0 ¶ µ µ c0 x0 b0 . ≤ ≤ u x l Since x¯00 is integer if and only if x0 and x are integral, and x¯0 is integer if and only if x00 and x are integral, than one can easily show that the system (2.1) has Chv´atal rank at most 1 if and only if the system (2.2) has Chv´atal rank at most 1. Since (2.2) has Chv´atal rank at most 1, also (2.1) has Chv´atal rank at most 1.

2.1 Classes of matrices with the Edmonds-Johnson property

2.1

17

Classes of matrices with the EdmondsJohnson property

There are few known classes of matrices known to have the Edmonds-Johnson property. In this section we briefly present the two main examples, while in Chapter 3 and 4 we introduce two new classes.

2.1.1

The Edmonds and Johnson’s class

Edmonds and Johnson [15, 16] derived the following theorem from Edmonds’ characterization of the matching polytope [13]. Theorem 2.2. P(Edmonds and Johnson [16].) If A = (αij ) is an integral matrix such that i |αij | ≤ 2 for each column index j, then A has the EdmondsJohnson property. We give a short proof of this result, essentially due to M. Singh [31]. Given a bidirected graph G and a subset F of its loops, we denote by A(G, F ) the matrix obtained from the sign matrix of G, Σ(G), by multiplying by 2 the columns corresponding to the loops in F . In what follows let A be the family of pairs (G, F ), where G is a bidirected graph and F is a subset of its loops. Thus Theorem 2.2, states that for each pair (G, F ) in A , A(G, F ) has the Edmonds-Johnson property. In the remaining of the section, we denote with A(G) the incidence matrix of a bidirected graph G. Notice that A(G) = A(G, L(G)). Next we show that, in proving Theorem 2.2, we can reduce ourselves to study systems of the form A(G) x = c (2.3) x ≥ 0, for every loopless bidirected graph G, and c ∈ ZV (G) . Notice that, if G is a loopless undirected graph, and c ∈ ZV (G), then the integer hull of (2.3) is the c-matching polytope. Lemma 2.3. If (2.3) has Chv´atal rank at most 1 for every loopless bidirected graph G and every integral c, then A(G, F ) has the Edmonds-Johnson property for every (G, F ) in A . Proof. At first we show that if (2.3) has Chv´atal rank at most 1 for every loopless bidirected graph G and every integral c, then the system A(G)x = c 0≤x≤u

(2.4)

18

The Edmonds-Johnson property

has Chv´atal rank at most 1 for every loopless bidirected graph G and every integral c, u. Let G be a loopless bidirected graph, let c, u be integral vectors and let x¯ be a vector in the first closure of (2.4). Now let G0 , c0 and x¯0 be obtained from G, c and x¯ in the following way. For each edge e = v1 v2 in E(G), add a new node ve and replace e with the path v1 , v1 ve , ve , ve v2 , v2 , such that v1 ve and ve v2 have a +1 sign in the vertex ve , the edge v1 ve has in v1 the same sign that e had in v1 , while the edge ve v2 has in v2 the opposite sign that e had in v2 , decrease c0v2 by σv2 ,e ue , set c0ve = ue , and set x¯0v1 ve = x¯e and x¯0ve v2 = ue − x¯e . Notice that also G0 is loopless. By Lemma 1.24 (i), x¯0 is in the first closure of A(G0 )x0 = c0 , x0 ≥ 0. If the latter system has Ch´atal rank at most 1, then x¯0 is a convex combination of integral solutions. Hence x¯ is a convex combination of integral vectors satisfying (2.4). Now we show that, if (2.4) has Chv´atal rank at most 1 for every loopless bidirected graph G and every integral c, u, then (2.4) has Chv´atal rank at most 1 for every bidirected graph G and every integral c, u. Let G be a bidirected graph, let c, u be integral vectors and let x¯ be a vector in the first closure of (2.4). Now let G0 be obtained from G by replacing every loop l incident with vl with two parallel edges l1 , l2 with endnodes vl , wl , where wl ∈ / V (G) and wl 6= wp , for every pair l, p of loops of G. Notice that 0 G is loopless. The sign of l1 and l2 in vl is the sign that l had in vl , l1 has in wl a +1 sign, and l2 has in wl a −1 sign. Let c0 be obtained from c by setting cwl = 0 for every loop l of G. Let x¯0 and u0 be obtained from x¯ and u by setting x¯0l1 = x¯0l2 = x¯l and u0l1 = u0l2 = ul for every loop l of G. Clearly x¯0 is in the first closure of A0 x0 = c0 , 0 ≤ x0 ≤ u0 . If the latter system has Ch´atal rank at most 1, then x¯0 is a convex combination of integral solutions. Hence x¯ is a convex combination of integral vectors satisfying (2.4). Now we show that if (2.4) has Chv´atal rank at most 1 for every bidirected graph G and every integral c, u, then the system b ≤ A(G)x ≤ c 0≤x≤u

(2.5)

has Chv´atal rank at most 1 for every bidirected graph G and every integral b, c, u. Let G be a bidirected graph, let b, c, u be integral vectors and let x¯ be a vector in the first closure of (2.5). Now let G0 be obtained from G by adding a new node v, a new edge ew = vw for every node w ∈ V (G), and a loop l incident with v. Let c0 be obtained from c by setting cv = c(V ).

2.1 Classes of matrices with the Edmonds-Johnson property

19

0

Let u0 ∈ ZE(G ) be obtained from u by setting u0l = +∞, and by setting u0ew = cw − bw for every new edge ew ∈ E(G0 ) \ (E(G) ∪ l). Let x¯0 be obtained from x¯ by setting x¯vw = (c − A(G)¯ x)w for every w ∈ V (G) and by 0 0 setting x¯l = b(V ). By Lemma 1.24 (i), x¯ is in the first closure of A0 x0 = c, 0 ≤ x0 ≤ u0 . If the latter system has Ch´atal rank at most 1, then x¯0 is a convex combination of integral solutions. Hence x¯ is a convex combination of integral vectors satisfying (2.5). Now we show that if (2.5) has Chv´atal rank at most 1 for every bidirected graph G and every integral b, c, u, then A(G) has the Edmonds-Johnson property for every bidirected graph G. Let b, c, l, u be integral vectors and let x¯ be a vector in the first closure of (2.3). Let b0 = b − Al, c0 = c − Al, u0 = u − l, x¯0 = x¯ − l. It is easy to see that x¯0 is in the first closure of b0 ≤ A(G)x0 ≤ c0 , 0 ≤ x0 ≤ u0 . If the latter system has Ch´atal rank at most 1, then x¯0 P is a convex combination of P Pintegral 0 i solutions, i.e. x¯ = i∈I λi y˜ , 0 ≤ λi ≤ 1, i∈I λi = 1. Hence x¯ = i∈I λi y i , where y i = y˜i + l is a convex combination of integral vectors satisfying (2.3). Finally we show that, if for every bidirected graph G, A(G) has the Edmonds-Johnson property, then for every pair (G, F ) in A , A(G, F ) has the Edmonds-Johnson property. Let (G, F ) be a pair in A , and let G0 be obtained from G by adding a new node v adjacent to each node of G incident with a loop in E(G) \ F , and by removing all the loops in E(G) \ F . Clearly, A(G, F ) is obtained from A(G0 ) by removing the row corresponding to the node v. If A(G0 ) has the Edmonds-Johnson property, then by Theorem 2.1 (iii), also A(G, F ) has the Edmonds-Johnson property. In the remaining of this thesis, whenever Z is a set, Y ⊆ Z, and z is a P Z vector in R , we denote with z(Y ) = i∈Y zi . Let G be a bidirected graph. It is well known that, for each U ⊆ V (G) such that U is connected in G, and c(U ) is odd, the inequality x(δ(U )) ≥ 1

(2.6)

is a Chv´atal inequality for the system (2.3). We call such constraint an oddcut inequality for (2.3). In the remainder of this section we will show that the polyhedron defined by the system A(G)x = c x(δ(U )) ≥ 1 x ≥ 0,

U ⊆ V (G), U connected, c(U ) odd

(2.7)

20

The Edmonds-Johnson property

is integral for every loopless bidirected graph G and every integral vector c. By Lemma 2.3, this proves Theorem 2.2. Now we give a lemma that uses standard uncrossing arguments (see [9, 22, 26, 27, 31] for details). This lemma is used in the proof of Theorem 2.2 and will also be used in Chapter 4. From now on, given a set W of vectors, we denote by spanW the space generated by the vectors in W . Whenever Z is a set, and Y ⊆ Z, we denote by χZ (Y ), or by χ(Y ) when there is no ambiguity, the vector in {0, 1}Z defined by χZ (Y )z = 1 if z ∈ Y , and χZ (Y )z = 0 otherwise. Lemma 2.4. (Uncrossing Lemma.) Let G = (V, E) be a graph, let c ∈ ZV , x¯ ∈ RE with x¯ > 0. Let F be the family of the subset U ⊆ V with c(U ) odd, such that x¯(δ(U )) = 1. Then there exists a laminar subfamily L of F such that span{χ(δ(U )) : U ∈ L } = span{χ(δ(U )) : U ∈ F }. Proof. Given a family F of subsets of V , we define spanF = span{χ(δ(U )) : U ∈ F }. Given two subsets T and S of V , we say that T and S intersect if the sets T ∩ S, T \ S, S \ T are all nonempty. Let F = {U ⊆ V : x¯(δ(U )) = 1, c(U ) odd}. Let L be a maximal independent laminar subfamily of F . We want to show that spanL = spanF . Otherwise there exists T ∈ F such that χ(δ(T )) ∈ / spanL . But then, since L is a maximal independent laminar subfamily of F , T intersects at least a set S ∈ L . Among all T ∈ F such that χ(δ(T )) ∈ / spanL , let T be one that intersects the minimum number of sets in L . Notice that, since c(T ) and c(S) are odd, then either c(S ∩ T ) and c(S ∪ T ) are odd, or c(S \ T ) and c(T \ S) are odd. In the first case, it can be checked that 2 = x¯(δ(S)) + x¯(δ(T )) ≥ x¯(δ(S ∩ T )) + x¯(δ(S ∪ T )) ≥ 2, hence both S ∩ T and S ∪ T are in F . As L is laminar, T ∩ S and T ∪ S intersect fewer sets from L than T . Hence by our choice of T , χ(δ(T ∩ S)), χ(δ(T ∪ S)) ∈ spanL . Since x¯ > 0, χ(δ(T )) = χ(δ(S ∩ T )) + χ(δ(S ∪ T )) − χ(δ(S)) is in spanL , a contradiction. In the second case, it can be checked that 2 = x¯(δ(S)) + x¯(δ(T )) ≥ x¯(δ(S \ T )) + x¯(δ(T \ S)) ≥ 2, hence both S \ T and T \ S are in F . As L is laminar, S \ T and T \ S intersect fewer sets from L than T . Hence by our choice of T , χ(δ(S \ T )), χ(δ(T \ S)) ∈ spanL . Since x¯ > 0, χ(δ(T )) = χ(δ(S\T ))+χ(δ(T \S))−χ(δ(S)) is in spanL , a contradiction. We say that a scalar x is half-integer if 2x is integer. Similarly, a vector or matrix A is called half-integral if 2A is integral.

2.1 Classes of matrices with the Edmonds-Johnson property

21

Proof of Theorem 2.2. By Lemma 2.3, we only need to show that system (2.7) defines an integral polyhedron for every loopless bidirected graph G and every integral vector c. By contradiction, suppose that there exists a loopless bidirected graph G and an integral vector c such that the system (2.7) has a fractional vertex x¯. Among all such counterexamples, choose G such that the pair (|V (G)|, |E(G)|) is lexicographically minimal. Let σ be the signing of G. Claim 2.1. G is connected. Proof of claim. If not, let G0 be a component of G such that x¯e ∈ / Z for some e ∈ E(G0 ). Let x¯0 , c0 be the restrictions of x¯, c, to E(G0 ). Notice that G0 is loopless and that |V (G0 )| < |V (G)|, hence the system (2.7) with respect to G0 , c0 has integral vertices. Then clearly x¯0 is a fractional vertex of the first closure of system (2.7) with respect to G0 , c0 , a contradiction. ¦ Notice that c(V (G)) is even, as otherwise the inequality x(δ(V (G))) ≥ 1 is not satisfied by x¯. Claim 2.2. x¯e > 0 for every e ∈ E(G). Proof of claim. If not, assume that there exists an edge e in E(G) with x¯e = 0. Let G0 be obtained from G by removing e. Let x¯0 be the vector obtained from x¯ by removing the component corresponding to e. Since G0 is loopless, |V (G0 )| = |V (G)|, and |E(G0 )| < |E(G)|, the system (2.7) with respect to G0 , c has integral vertices. Since the vector c has not changed, the odd-cut inequalities for A(G0 )x0 = c, x0 ≥ 0 are exactly the odd-cut inequalities for (2.7). Moreover notice that x¯0 (δ(U )) = x¯(δ(U )) for every U ⊆ V (G). Hence x¯0 is a fractional vertex of the system (2.7) with respect to G0 , c, a contradiction. ¦ Let F = {U ⊆ V | x¯(δ(U )) = 1}. By Claim 2.2, x¯ is the unique solution of the system A(G)x = c x(δ(U )) = 1

U ∈ F.

By Lemma 2.4, we can choose a laminar subfamily L of F such that x¯ is the unique solution of the system A(G)x = c x(δ(U )) = 1

U ∈ L,

(2.8)

and the elements of {χ(δ(U )) : U ∈ L } are not linear combination of rows of A(G). In particular, |E| ≤ |V | + |L |.

22

The Edmonds-Johnson property

Claim 2.3. G[S] is connected for every S ∈ L . Proof of claim. Otherwise, G[S] has a connected component T with c(V (T )) odd, thus δ(T ) ⊆ δ(S). So by Claim 2.2, 1 ≤ x¯(δ(T )) ≤ x¯(δ(S)) = 1. Therefore δ(T ) = δ(S), hence δ(S \ T ) = ∅, contradicting the fact that G is connected. ¦ Claim 2.4. L 6= ∅. Proof of claim. Otherwise |E| ≤ |V |. Thus G is a tree plus at most one edge. By Theorem 1.13, G is not bipartite, as otherwise A(G) is totally unimodular, and x¯ is integral, a contradiction. Thus G contains a cycle C, and C is odd. It follows that x¯e is half-integer for every edge e ∈ E(C). Since c is integral, then x¯e is integer for every e ∈ E(G) \ E(C), and, since x¯ is not integral, then x¯e is fractional for every e ∈ E(C). Since C is odd, there is an odd number of odd edges in C, thus c(V (G)) is odd, a contradiction. ¦ Let S be a minimal element of L . Claim 2.5. G[S] is not bipartite. Proof of claim. Otherwise, by Theorem 1.13, the nodes in G[S] can be partitioned into two subsets R, B such that, for every e = vw ∈ E(G), v and w are in the same class of the partition if and only if σv,e 6= σw,e . Since c(S) is odd, we may assume c(R) > c(B), thus c(R) − c(B) ≥ 1. Let R+ and R− (resp. B + and B − ) be the sets of edges vw ∈ δ(R) ∩ δ(S), v ∈ R (resp. vw ∈ δ(B) ∩ δ(S), v ∈ B), with respectively σv,vw = +1 and σv,vw = −1. P Then the odd-cut inequality relative to S, that is e∈R+ ∪R− ∪B + ∪B − xe ≥ 1, is satisfied tightly by x¯ and it is linearly independent from the equations in the system A(G)x = c. By summing together all the equalities in A(G)x = c corresponding to nodes in R and all the equalities in −A(G)x = −c corresponding to nodes in B we get X X xe − xe = c(R) − c(B). e∈R+ ∪B −

e∈R− ∪B +

P P P Hence 1 = P ¯e ≥ P ¯e − e∈R e∈R+ ∪R− ∪B + ∪B − x e∈R+ ∪B − x P − ∪B + x¯e ≥ 1, because x¯ ≥ 0. Thus e∈R+ ∪R− ∪B + ∪B − x¯e = e∈R+ ∪B − x¯e − e∈R− ∪B + x¯e , therefore R− ∪ B + = ∅, because x¯ > 0. But then the inequality defined by S is linearly dependent from Ax = c, a contradiction. ¦

2.1 Classes of matrices with the Edmonds-Johnson property

23

In particular, G[S] is not a tree. Now let G0 be the bidirected graph obtained from G by shrinking S into a new node s, where we identify each edge e = vw ∈ E(G) with v ∈ S and w ∈ / S with an edge e = sw in E(G0 ) (edges with both endnodes in S are removed), and where the signing σ 0 of 0 G is defined as follows. For every edge e = sw ∈ E(G0 ), let σs,e = +1 and 0 0 σw,e = σw,e , while for every edge e in E(G ) not incident with s and every 0 endnode v of e, let σv,e = σv,e . Let c0 be defined by c0v = cv for v ∈ V \ S and c0s = 1. Then the vector x¯0 obtained by restricting x¯ to the edges of G0 is a point that satisfies system (2.7) with respect to G0 , c0 . Furthermore, since G[S] is connected but not a tree, G0 has at most |E(G)| − |S| edges. Notice that x¯0 satisfies exactly all the equations in (2.8), except for the node equalities corresponding to the nodes in S. Thus, among this equalities, at least |E(G)| − |S| are linearly independent. Thus x¯0 is a vertex of the system (2.7) with respect to G0 , c0 , thus by induction it is integral. In particular |δ(S)| = 1 and, if f = st is the unique edge leaving S, x¯f = 1. Let G00 = G \ {f }. Now let x¯00 be the restriction of x¯ to the edges in E \ f . Let c00v = cv for v ∈ V \ {s, t}, c00s = cs − σs,st , c00t = ct − σt,st . We show that x¯00 satisfies the system (2.7) with respect to G00 , c00 . Clearly A(G00 )¯ x00 = c00 . 00 00 00 Let U ⊆ V (G ), such that U is connected in G , and c (U ) odd. Since G00 is not connected, then U ⊆ V (C), for a connected component C of G00 . Since s, t are in different connected components of G00 , by symmetry we assume s ∈ V (C). Thus, c00 (U ) is odd if and only either c(U ) is odd and s ∈ / U, 00 or if c(U ) is even and s ∈ U . In the former case, x¯ (δ(U )) = x¯(δ(U )) ≥ 1. In the latter case, c(C \ U ) is odd, thus x¯00 (δ(U )) = x¯(δ(C \ U )) ≥ 1. So x¯00 satisfies the system (2.7) with respect to G00 , c00 , hence by induction it is a convex combination of integral vectors y 1 , . . . , y k satisfying such system. µ ¶ x¯st Thus x¯ is a convex combination of , i = 1, . . . , k, which are integral yi vectors satisfying (2.7) with respect to G, c, a contradiction. ¤

2.1.2

The Gerards and Schrijver’s class

In this section we state a Theorem by Gerards and Schrijver [19], that characterizes Pthe Edmonds-Johnson property for integral matrices A = (αij ) that satisfy j |αij | ≤ 2 for each row index i. In Section 2.1.1 we showed that if A = (αij ) is an integral m × n-matrix such that m X |αij | ≤ 2 j = 1, . . . , n, (2.9) i=1

then A has the Edmonds-Johnson property.

24

The Edmonds-Johnson property

This property is not maintained when passing to transposes: (2.9) may not be replaced by n X |αij | ≤ 2 i = 1, . . . , m, (2.10) j=1

as the matrix

    M (K4 ) :=    

1 1 1 0 0 0

1 0 0 1 1 0

0 1 0 1 0 1

0 0 1 0 1 1

    ,   

the edge-node incidence matrix of the undirected graph K4 , does not have the Edmonds-Johnson property. Gerards and Schrijver proved that M (K4 ) is essentially the only counterexample among the matrices satisfying (2.10): Theorem 2.5. An integral matrix satisfying (2.10) has the Edmonds-Johnson property if and only if it cannot be transformed to M (K4 ) by a series of the following operations: (a) deleting or permuting rows or columns, or multiplying them by −1; µ ¶ 1 g (b) replacing matrix by the matrix D − f g, where f is a column f D vector and g a row vector. Notice that operation (b) corresponds to pivoting on a +1 entry, and then deleting the row and the column corresponding to the pivoted element. Notice that we can restrict ourselves to integral matrices A satisfying n X

|αij | = 2 (i = 1, . . . , m),

(2.11)

j=1

as rows with exactly one ±1 simply correspond to bounds on one variable. For every integral matrix A satisfying (2.11), let G(A) be the bidirected graph whose edge-node incidence matrix is A. Let σ be the signing of G(A). Clearly, the columns of A correspond to the nodes of G(A), and the rows to the edges. Let A0 be obtained by applying operation (b) in Theorem 2.5 to A. If the first row of A corresponds to an edge vw ∈ E(G(A)) with σv,vw 6= σw,vw , we get the ordinary graph contraction, thus G(A0 ) is obtained from G(A) by

2.1 Classes of matrices with the Edmonds-Johnson property

25

deleting the edge vw and identifying the nodes v and w. Otherwise, if the first row of A corresponds to an edge vw ∈ E(G(A)) with σv,vw = σw,vw , then G(A0 ) is obtained from G(A) by deleting the edge, changing sign to σv,vr for every node r incident with v, and identifying the nodes v and w. Thus we obtain the following equivalent form of Theorem 2.5. Corollary 2.6. A bidirected graph has the Edmonds-Johnson property if and only if it does not have a subgraph of the form

(2.12) where the wriggled lines stand for (pairwise openly disjoint) paths, such that each of the four cycles in (2.12) which have exactly three nodes of degree three, is odd. A graph as in (2.12) is called an odd-K4 . Gerards and Schrijver also described a smaller set of Chv´atal-Gomory cuts which are sufficient to give the convex hull of the integral solutions. Let A be matrix satisfying (2.11) with the Edmonds-Johnson property. Let G = G(A), V = V (G), and E = E(G). If x ∈ RV , b ∈ RE , e ∈ E and C is a subgraph of G, we denote with x(e) the entry in position e of Ax P (so x(e) = ±xv ± xw if e connects v and w), and we define x(C) := 1/2 e∈E(C) x(e), P and b(C) := e∈E(C) be . So Ax ≤ b is the same as: x(e) ≤ be for e ∈ E. If C is an odd cycle, the corresponding odd cycle inequality is: x(C) ≤

j1 2

k b(C) .

So it is a special type of Chv´atal-Gomory cut. In fact, for bidirected graphs, the odd cycle inequalities imply all other Chv´atal-Gomory cuts: Proposition 2.7. Let A be a matrix satisfying (2.11), let G = G(A) with node set V and edge set E, and let b ∈ ZE . Then the first closure of the system Ax ≤ b is given by the system Ax ≤ b x(C) ≤ b 12 b(C)c

C odd cycle of G.

26

2.2

The Edmonds-Johnson property

A conjecture by Gerards and Schrijver

Characterizing integral matrices with the Edmonds-Johnson property seems complicated. However, Gerards and Schrijver [18] noticed that there are some openings if we restrict ourselves to totally half-modular matrices, i.e. integral matrices in which each nonsingular square submatrix B has a halfintegral inverse. The reason seems that the ‘cuts’ HI determine an affine space over GF (2). Notice that the classes of matrices that we described in Sections 2.1.1 and 2.1.2 are totally half-modular, since it is well known that incidence matrices of bidirected graphs are totally half-modular. We now show that the operations that preserve the Edmonds-Johnson property, introduced in Theorem 2.1, also preserve total half-modularity. At first we give an easy technical lemma that will be used in the next proof and later in Chapter 4. Lemma 2.8. An m × n matrix A is totally half-modular if and only if for every nonsingular m × m submatrix B of (A I), B −1 is half-integral. Proof. Assume that for every nonsingular m × m submatrix B of (A I), B −1 is half-integral, and let A0 be a nonsingular square submatrix µ 0 of A. ¶ A 0 Now, for an identity matrix I of the appropriate size, the matrix 0 I is a nonsingular m × m submatrix of (A I), thus has half-integral inverse µ ¶ A0−1 0 . Hence A0−1 is half-integral, and A is totally half-modular. 0 I Conversely, assume that A is totally half-modular. Up to permuting rows 0 and columns, we may µ assume ¶ that each nonsingular m × m submatrix A of P 0 (A I) is of the form , where P is a nonsingular square submatrix of Q I A, thus has half-integral inverse by hypothesis, and where Q is a submatrix µ ¶ −1 P 0 of A and thus is integral. As A0 −1 = , P −1 is half-integral, −QP −1 I and Q is integral, then A0 −1 is half-integral.

Theorem 2.9. The class of totally half-modular matrices is closed under the following operations: (i) permuting rows and columns; (ii) multiplying rows and columns by −1;

2.2 A conjecture by Gerards and Schrijver

27

(iii) deleting rows and columns; (iv) dividing by k ∈ N, k ≥ 2 a row where all entries are multiple of k; µ

1 g f D

¶

(v) pivoting on a 1 entry, i.e. replacing matrix by the matrix µ ¶ −1 g , where f is a column vector and g a row vector. f D − fg Proof. Let A be a totally half-modular matrix. (i)-(iii): If A0 is obtained from A by permuting rows or columns, or by multiplying rows or columns by −1, or by deleting rows or columns, then trivially A0 is totally half-modular. (iv): Assume that A0 is obtained from A by dividing by k ∈ N, k ≥ 2 a row where all entries are multiple of k. By (i) we can assume that it is the first row of A. Let B 0 be a nonsingular square submatrix of A0 . Clearly we can assume that B 0 contains entries of first row of A, as otherwise B 0 is also a nonsingular square submatrix of A, and thus it has a half-integral inverse. Let B be the nonsingular square submatrix of A corresponding to B 0 . Clearly B 0 is obtained from B by dividing by k the entries of the first row of B. Thus B 0−1 is obtained from B −1 by multiplying by k the first column. Thus B 0−1 is half-integral because B −1 is half-integral and¶k ∈ N. µ 1 g is totally half(v): Assume that the m × n matrix A = f D modular. By Lemma submatrix B 0 of µ ¶ 2.8, for each m × m nonsingular µ ¶ 1 g 1 0 1 0 B= , B 0−1 is half-integral. Let U = , and notice f D 0 I −f I ¶ µ 1 g 1 0 that U is unimodular. Notice that U B = =: C. 0 D − f g −f I Now let C 0 be a m × m nonsingular submatrix of C. Clearly there exists an m × m nonsingular submatrix B 0 of B such that U B 0 = C 0 , thus C 0−1 = B 0−1 U −1 . Since U is unimodular, U −1 is integral, and since B 0−1 is halfintegral, it follows thatµC 0−1 is half-integral. Thus, by Lemma 2.8, and by ¶ −1 g (i) and (ii), the matrix is totally half-modular. f D − fg Notice that operations (i)-(v) of Theorem 2.9 are exactly operations (i)(v) of Theorem 2.1. Furthermore, since totally half-modular matrices have entries in {0, ±1, ±2}, then operation (iv) can be applied to such matrices only with k = 2.

28 We say that a matrix B operations (i)-(v). Let  2 1 A4 =  2 1 2 0

The Edmonds-Johnson property is a minor of A if it arises from A by a series  1 0 0 1 , 1 1

 1 2 0 A3 =  1 2 2  . 1 0 2 

Gerards and Schrijver [18] formulated the following conjecture: Conjecture 2.1. A totally half-modular matrix A has the Edmonds-Johnson property, if and only if it has no minor equal to A4 or A3 . Next we observe that the matrices A4 and A3 do not have the EdmondsJohnson property. To this aim, we need the following technical lemma, that will also be needed in Chapter 4. Lemma 2.10. If A is a totally half-modular matrix and b is an integral vector, any nontrivial Chv´atal inequality for Ax ≥ b is equivalent to an inequality of the form (λA)x ≥ dλbe such that λ has only 0, 21 entries, λA is integral, λb is not integral, and the positive components of λ correspond to linearly independent rows of A. Proof. By Lemma 1.20, the first Chv´atal closure of Ax ≥ b is obtained by adding the inequalities (λA)x ≥ dλbe for all vectors 0 ≤ λ < 1 such that λA is integral and λb is not integral. By Caratheodory’s theorem, we may assume that the positive components of λ correspond to linearly independent rows of A. As each nonsingular square submatrix of A has half-integral inverse, it follows that λ is half-integral. Observation 2.11. The matrices A4 and A3 do not have the EdmondsJohnson property. Proof. Consider the system A4 x ≥ 1, x ≥ 0, and let x¯i = 1/3 for i = 1, . . . , 4. By Lemma 2.10, the nontrivial  1 1 1  1 1 0   1 0 1 3 1 1

(2.13)

Chv´atal inequalities for (2.13) are    0 1  1  1  x ≥  . (2.14)  1  1  1 2

2.2 A conjecture by Gerards and Schrijver

29

Notice that x¯ satisfies system (2.13), and satisfies tightly all the four Chv´atal inequalities in (2.14), that are linearly independent. Thus x¯ is a fractional vertex of the first closure of (2.13). Consider the system 

 1 A3 x ≥  2  , x ≥ 0, 1

(2.15)

and let x¯i = 1/2 for i = 1, 2, 3. By Lemma 2.10, the nontrivial Chv´atal inequalities for (2.15) are     1 1 0 1  1 0 1   1      (2.16)  1 2 1 x ≥  2 . 1 1 2 2 Notice that x¯ satisfies system (2.15), and satisfies tightly all the Chv´atal inequalities in (2.16). Since the constraint matrix of (2.16) has rank 3, then x¯ is a fractional vertex of the first closure of (2.15). Notice that, in the class of matrices studied by Edmonds and Johnson (see Section 2.1.1), the matrices A4 and A3 do not appear. In the class of matrices studied by Gerards and Schrijver (see Section 2.1.2) only A4 appears. In fact, notice that A4 can be obtained from M (K4 ) by pivoting the 1 entries in positions (1, 2), (2, 3), and (3, 4), by multiplying by −1 the first column of the obtained matrix, and then by removing the first three rows. In the remaining two chapters we present our contributions. In Chapter 3, we study systems of the from b ≤ Mx ≤ c l ≤ x ≤ u,

(2.17)

for integral vectors b, c, l, u, where M is obtained from a totally unimodular matrix with two nonzero elements per row by multiplying by 2 some of its columns. The case where M is obtained from the transpose of the incidence matrix of a bipartite graph by multiplying by 2 some of the columns, has been studied by Conforti et al. in [7]. In this case, they derived an explicit characterization of the inequalities defining the integer hull.

30

The Edmonds-Johnson property

We give an explicit description of a totally dual integral system that describes the integer hull of the polyhedron P defined by (2.17). Since the inequalities of such totally dual integral system are Chv´atal inequalities for P , this implies that the matrix M has the Edmonds-Johnson property. Thus also in this class, the matrices A4 and A3 do not appear. The results in Chapter 3 are joint work with G. Zambelli [11]. In Chapter 4 we study totally half-modular matrices with entries in {0, ±1, ±2} obtained from sign matrices of bidirected graphs by multiplying by 2 some of the columns. We show that the matrix   1 1 2 0 (2.18) M4 =  −1 0 0 2  0 1 0 2 is the only minor minimal matrix in such class (up to multiplying rows and columns by −1) that does not have the Edmonds-Johnson property. Notice that this result extends Theorem 2.2 of Edmonds and Johnson. The results in Chapter 4 are joint work with A. Musitelli and G. Zambelli [10]. A partial result was shown by Del Pia and Zambelli [12].

Chapter 3 Bipartite vertex covers with parity conditions In this chapter we study systems of the from b ≤ M x ≤ d, l ≤ x ≤ u, where M is obtained from a totally unimodular matrix with two nonzero elements per row by multiplying by 2 some of its columns, and b, d, l, u are integral vectors. We give an explicit description of a totally dual integral system that describes the integer hull of the polyhedron P defined by the above inequalities. Since the inequalities of such totally dual integral system are Chv´atal inequalities for P , our result implies that the matrix M has the Edmonds-Johnson property. We also derive a strongly polynomial time algorithm to find an integral optimal solution for the dual of the problem of maximizing a linear function with integer coefficients over the aforementioned totally dual integral system.

3.1

Introduction

Let M be a matrix obtained from a totally unimodular matrix with exactly two nonzero elements in every row by multiplying by 2 some of its columns, and let b, c, d, l, u be integral vectors of appropriate dimension. We consider problems of the form max c> x b ≤ M x ≤ d, l ≤ x ≤ u, x integral.

(3.1)

The case where M is obtained from the transpose of the incidence matrix of a bipartite graph by multiplying by 2 some of the columns, and where

32

Bipartite vertex covers with parity conditions

the variables are restricted to be nonnegative, has been studied by Conforti et al. in [7]. In this case, they showed that the above problem can be solved in strongly polynomial time. This was accomplished by expressing the integer hull of the system as the projection of some polyhedron in a higher dimensional space. From such extended formulation, they derived an explicit characterization of the inequalities defining the integer hull. In this chapter we show, with a similar construction, how problem (3.1) can in fact be reduced to a weighted vertex covering problem on a certain extended graph. Using this construction, we describe a totally dual integral system defining the integer hull of the polyhedron defined by the constraint system of (3.1). µ ¶ M Note that, since is also obtained from a totally unimodular matrix −M with two nonzero elements per row by multiplying some of its columns by 2, it is sufficient to consider systems of the form

l≤

Mx ≥ b x ≤ u.

(3.2)

The matrix M can be represented as a bidirected graph G = (V, E, σ): the columns of M correspond to the nodes of G, and the rows to the edges. A row corresponds to an edge e connecting the two nodes where the nonzero elements occur, with σv,e = +1 if Me,v ∈ {+1, +2}, with σv,e = −1 if Me,v ∈ {−1, −2}, and with σv,e = 0 otherwise. Let I be the set of nodes of V corresponding to the columns of M with entries 0, ±2. Let L = V \ I. Thus M is obtained from the edge-node incidence matrix of G by multiplying by 2 the columns corresponding to nodes in I. For every bidirected edge e ∈ E we call be the requirement of e. An I-trail in G is a trail T = v1 , . . . , vk in G such that v1 ∈ I, v2 , . . . , vk−1 ∈ / I. An I-path in G is an I-trail in G which is a path in G. For any such I-trail in G we define γ1T = σv1 ,v1 v2 ; γiT = (σvi ,vi−1 vi + σvi ,vi vi+1 )/2; i = 2, . . . , k − 1 γkT = σvk ,vk−1 vk . Notice that γiT ∈ {0, ±1} for every i = 1, . . . , k, and that it is possible that vs = vt and γsT 6= γtT for two indices s 6= t. Given an I-trail P = v1 , . . . , vk of G, the following inequalities are Chv´atal-Gomory inequalities for (3.2), thus they are valid for its integer hull:

3.1 Introduction

Pk i=1

γiP xvi ≥

i=1

γiP xvi ≥

i=1

γiP xvi ≥

i=1

γiP xvi ≥

i=1

γiP xvi ≥

Pk

Pk−1 Pk−1 Pk

33

lP

be

m

if v1 , vk ∈ I, m  e∈P be +lvk  m lP 2 if vk ∈ / I, γkP = 1, e∈P be −uvk  lP 2 m  e∈P be +lvk  lP 2 m if vk ∈ / I, γkP = −1. e∈P be −uvk  e∈P

lP

2

(3.3)

2

In fact, all such inequalities are obtained by summing up the inequalities of M x ≥ b corresponding to the edges of the I-trail, plus or minus the lower or upper bound on the variable corresponding to endnodes not in I, dividing the inequality thus obtained by 2, and rounding up the right-hand-side. Theorem 3.1. The system defined by (3.2) and (3.3), for every I-path P , is totally dual integral. Notice that, in Theorem 3.1 we only need inequalities of the form (3.3) when P is an I-path, rather than a general I-trail. We postpone the proof to Section 3.2.2. Our proof yields a strongly polynomial time algorithm that, given an integral cost vector c, finds an integral optimal solution for the dual of {max c> x : x satisfies (3.2), (3.3) for every I-path P } whenever such problem has a finite optimum. Deriving a polynomial algorithm from the proof, however, is non-trivial, and it is accomplished in Section 3.4. Edmonds and Giles [14] showed that if a system of linear inequalities with integer coefficients is totally dual integral, then the polyhedron defined by such a system is integral (cf. Corollary 1.16). Thus the above theorem implies the following. Corollary 3.2. The polyhedron defined by (3.2) and (3.3), for every I-path P , is integral. Since the inequalities in (3.3) are Chv´atal inequalities for (3.2), Theorem 3.1 implies the following. Corollary 3.3. Let M be a matrix obtained from a totally unimodular matrix with two nonzero elements per row by multiplying some of its columns by 2. Then M has the Edmonds-Johnson property.

34

Bipartite vertex covers with parity conditions

3.2

Bipartite case

In this section we study a special case of problem (3.2), namely the case where M is obtained from a nonnegative totally unimodular matrix with two nonzero elements per row by multiplying by 2 some of its columns, and where all the variables are required to be nonnegative (i.e., l = 0, u = +∞). In Section 3.3 we will see how the general case reduces to this simpler case. In this special case the matrix M can be represented as an undirected graph G = (V, E): the columns of M correspond to the nodes of G, and the rows to the edges. A row corresponds to an edge connecting the two nodes where the nonzero elements occur. By Theorem 1.11, G is bipartite. Let U, W be the sides of G. For every edge e ∈ E we call be the requirement of e. For every i ∈ U ∪ W we define ai = 2 if i ∈ I, ai = 1 if i ∈ L. Hence, given a cost vector c ∈ ZU ∪W , we consider the following problem: P min i∈U ∪W ci xi s.t. ai xi + aj xj ≥ bij ij ∈ E (3.4) xi ≥ 0 i∈U ∪W xi ∈ Z i ∈ U ∪ W. We will show the following. Theorem 3.4. The system of linear inequalities a i xi + a j xj ≥ l bij m P b(P ) i∈P xi ≥ 2 xi ≥ 0

ij ∈ E P I-path i∈U ∪W

(3.5)

is totally dual integral. To show Theorem 3.4 we need to show that the problem P l b(P ) m P b y + max yP e e Pe∈E PP 2 i∈U ∪W s.t. P 3i yP ≤ ci e3i ai ye + ye ≥ 0 e∈E yP ≥ 0 P I-path,

(3.6)

has an integral optimal solution y for each vector c ∈ ZU ∪W for which min{c> x : x satisfies (3.5)} has a finite optimum. Since the latter problem is unbounded whenever c has a negative component, throughout this section we will assume that c is nonnegative. We show how this problem can be reduced to a problem where the constraint matrix is the transpose of the incidence matrix of some extended bipartite graph.

3.2 Bipartite case

3.2.1

35

The extended graph

Given a bipartite graph G = (U ∪ W, E), and a requirement vector b ∈ ZE , ˜ ∅,b = (U˜∅ ∪ W ˜ ∅ , E) ˜ as follows. we define G Let U 0 , W 0 be copies of U , W , respectively, such that U , W , U 0 , W 0 are pairwise disjoint. For every i ∈ U ∪ W , we denote by i0 the copy of i in ˜ ∅ = W ∪ W 0 . E˜ contains the edges ij and i0 j 0 for U 0 ∪ W 0 . Let U˜∅ = U ∪ U 0 , W every ij ∈ E such that bij is odd, and the edges i0 j and ij 0 for every ij ∈ E such that bij is even. ˜ I,b = (U˜I ∪ W ˜ I , E) ˜ is obtained from For I ⊆ U ∪ W , the extended graph G 0 ˜ ∅,b by identifying the two copies i, i of every node i ∈ I, where U˜I and W ˜I G ˜ ˜ ˜ correspond to U∅ and W∅ . (Notice that we identify the set of edges of G∅,b ˜ I,b .) with that of G ˜ I,b are the nodes i, i0 (where For each node i ∈ U ∪ W , the images of i in G 0 ˜ I,b are the edges i = i if i ∈ I). For each edge ij ∈ E, the images of ij in G 0 0 0 0 ij and i j if bij is odd, the edges ij and i j if bij is even. We say that a ˜ I is the symmetric of another node j ∈ U˜I ∪ W ˜ I , and we node i ∈ U˜I ∪ W write i = sym(j), when i, j are the images (possibly coincident) of the same element of U ∪ W . We say that an edge e1 ∈ E˜ is the symmetric of another ˜ and we write e1 = sym(e2 ), when the two edges e1 and e2 are edge e2 ∈ E, distinct images of the same edge e ∈ E. ˜be as follows. For each edge To each edge e in E˜ we assign a requirement ¥ bij ¦ §b ¨ ij ∈ E with bij odd we define ˜bij = 2 and ˜bi0 j 0 = 2ij , while for every b edge ij ∈ E such that bij is even we define ˜bi0 j = ˜bij 0 = 2ij . ˜ I we assign a cost c˜w equal to the cost of its To each node w of U˜I ∪ W corresponding node in U ∪ W . ˜ I,b Now consider the following problem on G min s.t.

P

c˜i x˜i x˜i + x˜j ≥ ˜bij x˜i ≥ 0

ij ∈ E˜ ˜I i ∈ U˜I ∪ W

(3.7)

P ˜b y˜ max P e∈E˜ e e s.t. ˜e ≤ c˜i e3i y y˜e ≥ 0

˜I i ∈ U˜I ∪ W ˜ e ∈ E.

(3.8)

˜I ∪W ˜I i∈U

and its dual problem

Note that the constraint matrix of (3.8) is the incidence matrix of a bipartite graph, and thus it is totally unimodular by Theorem 1.11. Thus, if problems (3.7) and (3.8) admit optimal solutions, then they admit optimal solutions

36

Bipartite vertex covers with parity conditions

that are integral, provided that ˜b and c˜ are integral. The LP (3.7) is similar to an extended formulation introduced in [6] and used also in [7]. Note that, ˜ I,b , if x is a feasible solution for (3.4) then by construction of G x˜i = x ¥ ixi ¦ i ∈ I x˜i = § 2 ¨ i ∈ (U ∪ W ) \ I x˜i0 = x2i i0 ∈ (U 0 ∪ W 0 ) \ I is a feasible integral solution for (3.7) with the same objective value. If x ˜ is an integral feasible solution for (3.7) then xi = x˜i i∈I 0 xi = x˜i + x˜i i ∈ L is a feasible solution for (3.4) with the same objective value. Hence (3.4), (3.7) and (3.8) have the same optimal value. Also, since any feasible solution of (3.4) is feasible for (3.5), then by weak duality the optimal value of (3.6) is at most the optimal value of (3.4), and therefore of (3.8).

3.2.2

Proof of Theorem 3.4

We prove Theorem 3.4 by showing how to derive an integral optimal solution for (3.6) from an integral optimal solution for (3.8). First, we need to prove a lemma. ˜ is an antisymmetric orientation of G ˜ I,b if D ˜ is We say that a digraph D ˜ obtained from GI,b by orienting its edges so that, for any pair of symmetric ˜ I,b , one of the two is oriented from U˜I to W ˜ I , and the other edges e1 , e2 of G ˜ I to U˜I . For ease of notation, in the remainder, whenever we refer to from W ˜ I,b , we also denote by e the arc of D ˜ obtained by orienting e. an edge e of G ˜ I as β(u,v) = ˜buv and We define the cost of each arc (u, v) from U˜I to W ˜ I to U˜I as β(v,u) = −˜bvu . Given a directed the cost of each arc (v, u) from W ˜ we define the cost of S as β(S). path or cycle S in D, Given a directed path (resp. a directed cycle) S = v1 , v2 , . . . , vn in ˜ D, sym(S) = sym(vn ), sym(vn−1 ), . . . , sym(v2 ), sym(v1 ) is a directed path ˜ We (resp. a directed cycle), which we refer to as the symmetric of S in D. ˜ say that a directed path or a directed cycle in D is symmetric if it coincides with its symmetric. ˜ is symmetric if and only if S Observation 3.5. A directed path S in D contains exactly one node i ∈ I and S = Q, i, sym(Q) for some directed path ˜ that ends in i. Q in D A directed cycle S is symmetric if and only if S contains exactly two distinct

3.2 Bipartite case

37

˜ from nodes i, j ∈ I and S = i, Q, j, sym(Q), i for some directed path Q in D i to j.

v3

v4

v5 ...

v2

v1 = sym(v1)

vk = sym(vk)

...

sym(v2) sym(v3)

sym(v4)

sym(v5)

Figure 3.1: a symmetric directed cycle. ˜ then S 0 = Proof. If S = v1 , v2 , . . . , vn is a symmetric directed path in D, v2 , . . . , vn−1 is also symmetric, so by induction on the length of S we may assume that S 0 contains exactly one node i ∈ I and S 0 = Q0 , i, sym(Q0 ) for ˜ that ends in i. Since (vn−1 , vn ) = sym(v1 , v2 ), if some directed path Q0 in D we define Q = v1 , v2 , Q0 , i, then S = Q, i, sym(Q). Let S be a symmetric directed cycle. If S does not contain any node in L, then S consists of two distinct nodes i, j ∈ I and of the two symmetric edges (i, j) and (j, i). So we may assume that S contains a node w ∈ / I and its symmetric. Since the two distinct paths in S with endnodes w and sym(w) are both symmetric, the statement follows from the case of the symmetric directed path. ˜ be an antisymmetric orientation of G ˜ ∅,b . Given a diLemma 3.6. Let D ˜ then −1 ≤ β(S) + β(sym(S)) ≤ 1, where β(S) + rected path S in D, β(sym(S)) = 0 if and only if the endnodes of S are either both in U ∪ W 0 or ˜ then β(S) + β(sym(S)) = 0. both in U 0 ∪ W . Given a directed cycle S in D, ˜ One can readily Proof. Let S be a directed path or a directed cycle in D. verify that β(S) + β(sym(S)) = −|{(u, v) ∈ S : u ∈ U, v ∈ W }| +|{(v, u) ∈ S : v ∈ W, u ∈ U }| +|{(u0 , v 0 ) ∈ S : u0 ∈ U 0 , v 0 ∈ W 0 }| −|{(v 0 , u0 ) ∈ S : v 0 ∈ W 0 , u0 ∈ U 0 }|.

38

Bipartite vertex covers with parity conditions

The arcs leaving U ∪ W 0 are either arcs from U to W or from W 0 to U 0 , while the arcs entering U ∪ W 0 are either arcs from W to U or from U 0 to W 0 . Therefore, by the above equation, β(S) + β(sym(S)) is the difference between the number of arcs in S entering U ∪ W 0 and the number of arcs in S leaving U ∪ W 0 . Since S is a directed path or a directed cycle, the absolute value of this difference is at most 1, and it is 0 if and only if S is a directed path with endnodes either both in U ∪ W 0 or both in U 0 ∪ W , or if S is a directed cycle. Proof of Theorem 3.4. ˜ I,b = (U˜I ∪ W ˜ I , E) ˜ simply by G ˜ = (U˜ ∪ Through this proof, we denote G ˜ ˜ W , E). Let y˜ be an integral optimal solution of (3.8). We will show how to derive from y˜ an integral solution y for (3.6) with value ˜b> y˜, thus showing that y is optimal for (3.6). We say that an edge e ∈ E˜ is loaded (for y˜) if y˜e > 0, unloaded (for y˜) if y˜e = 0. P We prove the theorem by induction on i∈U ∪W ci . If E contains an edge e such that both its images in E˜ are unloaded, and if y 0 is an integral optimal solution for the instance of (3.6) on the graph G0 = (U ∪ W, E \ {e}) with value ˜b> y˜, then the vector y, obtained by completing y 0 with ye = 0 and with yP = 0 for every I-path P in G that contains e, is an integral optimal solution for the problem (3.6) with value ˜b> y˜. Thus from now on we will assume that for every e ∈ E, at least one of its two images in E˜ is loaded. We will use two types of reductions, defined next. Reduction (with respect to y˜) on the symmetric edges. ˜ contains a pair of symmetric edges e1 and e2 that are both loaded. Suppose G ˜ let γe = min{˜ For any edge e ∈ E with images e1 , e2 ∈ E, ye1 , y˜e2 }, and define 0 a new cost vector c on the nodes of G by X c0i = ci − ai γij , i ∈ U ∪ W. ij∈E

We call reduced problem (w.r.t. y˜) the instance of (3.6) on the graph G with costs on the nodes c0 , and extended reduced problem (w.r.t. y˜) the corresponding instance of (3.8). Notice that the vector y˜0 defined by y˜e0 1 = y˜e1 − γe y˜e0 2 = y˜e2 − γe .

e∈E

is an integral optimal solution to the extended reduced problem, with value ˜b> y˜ − P e∈E be γe .

3.2 Bipartite case

39

P P Since i∈U ∪W c0i < i∈U ∪W ci , by induction the reduced problem has an integral optimal solution y 0 with value ˜b> y˜0 . Hence the vector y defined by ye = ye0 + γe e ∈ E yP = yP0 P I-path, is a feasible integral solution for problem (3.6) with value ˜b> y˜, thus y is optimal. Thus we may assume that for every edge e ∈ E˜ exactly one among e ˜ be the digraph obtained from G ˜ by orienting and sym(e) is loaded. Let D ˜ ˜ ˜ ˜ from U to W the unloaded edges, and from W to U the loaded edges. Note ˜ is an antisymmetric orientation of G. ˜ We denote by D ˜ ∅,b the digraph that D ˜ ∅,b by orienting the unloaded edges of E˜ from U ∪ U 0 to obtained from G ˜ ∅,b W ∪ W 0 and the loaded edges of E˜ from W ∪ W 0 to U ∪ U 0 . Notice that D ˜ ∅,b , and that D ˜ can be obtained from is an antisymmetric orientation of G ˜ ∅,b by identifying the images of nodes in I. D Next we define the second type of reduction. Reduction on a symmetric path or symmetric cycle of non-positive cost. ˜ of non-positive cost, or a symmetric Let S be a symmetric directed cycle of D ˜ of non-positive cost with c˜k − P y˜e > 0, where k is directed path of D e3k the only endnode of S incident with an unloaded arc of S. In the first case let γ = min{˜ ye : e ∈ S, y˜e > 0}, in the second one let γ be the minimum P among c˜k − e3k y˜e , and min{˜ ye : e ∈ S, y˜e > 0}. Let c0 be the cost vector on the nodes of G obtained from c by setting c0i = ci for any node i whose images are not in S, and c0i = ci − γ for any node i whose images are in S. We call reduced problem the instance of (3.6) on the graph G with costs c0 , and extended reduced problem the corresponding instance of (3.8). Then y˜e0 = y˜e − γ e ∈ S, y˜e > 0 y˜e0 = y˜e otherwise is an integral optimal solution for the extended reduced problem. By Observation 3.5, S is the union of a path Q and its symmetric sym(Q), and the endnodes in common among Q and sym(Q) are the only nodes of S in I. Note that the set of edges of G whose images are in S define an I-path R in G. If S is a path, then the unique node of S in I is also the unique endnode of R in I, while if S is a cycle, then the two distinct nodes of S in I are the endnodes of R. Notice that X X X X X ˜be = ˜be + ˜be = β(S) + 2 ˜be . be = (3.9) e∈R

e∈S

e∈S e loaded

e∈S e unloaded

e∈S e loaded

40

Bipartite vertex covers with parity conditions

The arcs of Q and of sym(Q) induce directed paths Q0 and sym(Q0 ) in ˜ ∅,b , respectively. Furthermore β(S) = β(Q0 ) + β(sym(Q0 )). By Lemma 3.6 D |β(Q0 ) + β(sym(Q0 ))| ≤ 1. Since S has non-positive cost, β(S) ∈ {0, −1}. Thus, by (3.9), » ¼ X ˜be = b(R) . 2 e∈S e loaded

¨ § ˜b> y˜− b(R) γ. Hence the optimal value of the extended reduced problem is 2 P P Since i∈U ∪W c0i < i∈U ∪W ci , by induction there exists an integral solution y 0 for the reduced problem with value ˜b> y˜0 , thus the vector y defined by ye = ye0 e∈E yR = yR0 + γ yP = yP0 P I-path, P 6= R is an integral feasible solution for problem (3.6) with value ˜b> y˜, hence y is optimal. ˜ as the elements of {u ∈ U˜ : P y˜e < We define the sources of D e3u ˜ as the elements of {u ∈ U˜ : ˜ : P y˜e > 0} and the sinks of D c˜u } ∪ {v ∈ W e3v P ˜ : P y˜e < c˜v }. Let S be either a directed path in y ˜ > 0} ∪ {v ∈W e e3v e3u ˜ from a source to a sink or a directed cycle, and ε be a positive number. We D say that the solution y˜0 is obtained by augmenting y˜ by ε on S if y˜e0 = y˜e + ε for every unloaded edge e ∈ S, y˜e0 = y˜e − ε for every loaded edge e ∈ S, and y˜e0 = y˜e for every edge e ∈ E˜ \ S. If ε is small enough, then y˜0 is also a feasible solution for problem (3.8), with value ˜b> y˜ + εβ(S) (notice that this is the standard notion of augmentation in flow theory, see for example [30]). Therefore, since y˜ is an optimal solution, we have the following. Observation 3.7. If S is a directed path from a source to a sink or a directed ˜ then β(S) ≤ 0. Furthermore, if β(S) = 0, then for ε > 0 small cycle in D, enough the solution obtained by augmenting y˜ by ε on S is optimal for (3.8). ˜ contains a directed cycle S. We show that in this Suppose now that D ˜ contains a directed cycle C that either is symmetric or has at most one case D node in I. In fact, if S contains two or more nodes in I, let Q be a minimal directed path contained in S with endnodes in I and with no intermediate node in I. The directed graph induced by the arcs of Q∪sym(Q) is the union of arc-disjoint directed cycles, so it either contains a directed cycle with at most one node in I, or it is a symmetric directed cycle. ˜ Case 1: C is a symmetric directed cycle in D.

3.2 Bipartite case

41

By Observation 3.7, β(C) ≤ 0, thus we can apply the reduction on the symmetric directed cycle C of non-positive cost, and we are done. Case 2: C has at most one node in I. ˜ ∅,b , The arcs in C form a directed cycle or a directed path in the digraph D thus, by Lemma 3.6, −1 ≤ β(C) + β(sym(C)) ≤ 1, while by Observation 3.7 β(C) ≤ 0 and β(sym(C)) ≤ 0. Hence at least one among C and sym(C) has cost zero, and we assume β(C) = 0. Note that C must cross an unloaded arc e¯ whose symmetric is not in C, otherwise all the unloaded arcs of C have their symmetric in C, thus C is symmetric. So we can augment y˜ by min{˜ ye : e ∈ C, y˜e > 0} on C thus getting another integral optimal solution 0 y˜ where both e¯ and its symmetric have strictly positive value. Thus we can now apply the reduction w.r.t. y˜0 on the symmetric edges, and we are done. ˜ is acyclic. Notice that every Hence we can assume that the digraph D ˜ with in-degree 0 is a source. In fact if j has in-degree node not isolated in D 0 and strictly positive out-degree, then sym(j) has out-degree P 0 and strictly P ˜ positive in-degree. So, if j ∈ U , then e3j y˜e = 0 and e3sym(j) y˜e > 0, ˜ , then P y˜e > 0. In the same way notice that every node not if j ∈ W e3j ˜ with out-degree 0 is a sink. isolated in D ˜ Since Suppose that there exists a node i in I that is not isolated in D. ˜ is acyclic, there exists a path Q from i to a node j of out-degree 0 D ˜ and, since j has out-degree 0, j ∈ in D / I. Consider the directed walk ˜ is acyclic, S must be a S = sym(j), sym(Q), i, Q, j. Notice that, since D ˜ from a source to a sink, by directed path. Since S is a directed path in D Observation 3.7 β(S) ≤ 0. Moreover, ifPk is the only endnode of S incident with an unloaded arc of S, then c˜k − e3k y˜e > 0, thus we may apply the reduction on the symmetric directed path of non-positive cost S, and we are done. ˜ So we can assume that all the nodes in I are isolated in D. ˜ Therefore there exists a directed path S in D from a node with in-degree 0 to a node with out-degree 0. Since both S and sym(S) are directed paths in ˜ from a source to a sink, by Observation 3.7 β(S) ≤ 0 and β(sym(S)) ≤ 0. D By Lemma 3.6, −1 ≤ β(S) + β(sym(S)) ≤ 1. Hence at least one among S and sym(S) has cost zero, and we assume it is S = v1 , . . . , vk . Notice that S crosses an unloaded arc e¯ whose symmetric is not in S. In fact, if all the unloaded arcs of S have their symmetric in S, it must be |e ∈ S : e loaded| = |e ∈ S : e unloaded| + 1, since in S unloaded and loaded arcs alternate and since S is not symmetric. But then (v1 , v2 ) and (vk−1 , vk ) are both loaded and at least one of them is the symmetric of an unloaded arc in S. By symmetry we may assume it is (v1 , v2 ), thus sym(v1 ) has out-degree 0, hence

42

Bipartite vertex covers with parity conditions

(sym(v2 ), sym(v1 )) is the last arc of S. A contradiction as it is unloaded. So we can augment y˜ on S by the minimum among c˜j for every endnode j of S incident with an unloaded arc of S, and min{˜ ye : e ∈ S, y˜e > 0}. Thus 0 we get another integral optimal solution y˜ where both e¯ and its symmetric have strictly positive value. Hence we can now apply the reduction w.r.t. y˜0 on the symmetric edges, and we are done. ¤ We conclude the section with the following corollary, which will be used in the proof of Theorem 3.1. Corollary 3.8. Let l ∈ ZU ∪W . The system ai xi + aj xj xi P i∈P xi P i∈P xi

≥ bij ≥ lli m ) ≥ b(P l 2 m b(P )+lvk ≥ 2

ij ∈ E i∈U ∪W P = v1 , . . . , vk I-path, v1 , vk ∈ I

(3.10)

P = v1 , . . . , vk I-path, vk ∈ / I.

is totally dual integral. Proof. Let c be a vector in ZU ∪W . By Theorem 3.4 we know that the dual of the problem min{c> x : x satisfies (3.5)} with integer requirements bij − ai li − aj lj has an integral optimal solution y ∗ . It is straightforward to check that the integral solution y¯ defined by y¯e = ye∗ , e ∈ E, y¯P = yP∗ , P P P I-path, y¯i = ci − e3i ai ye∗ − P 3i yP∗ , i ∈ U ∪ W , is optimal for min{c> x : x satisfies (3.10)}. ¤

3.3

General case

In this section we prove Theorem 3.1 by reducing the general problem to the bipartite case studied in Section 3.2. Proof of Theorem 3.1. First we show the following. Claim. The system defined by (3.2) and (3.3), for every I-trail P , is totally dual integral. Proof of claim. We show how to reduce this problem to the previous case. We define the undirected graph G0 = (V ∪V 0 , E 0 ) as follows. Let V 0 be a copy of V such that V ∩ V 0 = ∅. For every i ∈ V we denote by i0 the copy of i in V 0 and for every X ⊆ V we denote by X 0 the subset of V 0 that contains only the copies of the nodes in X. E 0 contains the edge ii0 for every i ∈ V , with

3.3 General case

43

requirement 0, and the edge ij (resp. ij 0 , i0 j, i0 j 0 ) for every edge ij ∈ E with σi,ij = σj,ij = +1 (resp. σi,ij = +1 and σj,ij = −1, σi,ij = −1 and σj,ij = +1, σi,ij = σj,ij = −1), with the same requirement of the original bidirected edge 0 ij. Let b0 ∈ ZE be the vector of requirements on the edges in E 0 . Since the edge-node incidence matrix of G is totally unimodular and has two nonzero elements per row, it follows from Theorem 1.13 that V can be partitioned into two sets R, B such that every edge of G with the same sign in both its endnodes has one endnode in R and the other in B, while every edge with different signs in its endnodes is contained in R or B. Therefore every edge of G0 has exactly one endnode in R ∪ B 0 and the other in R0 ∪ B, thus G0 is bipartite. If we define a0i = 2 for i ∈ I ∪ I 0 and a0i = 1 for i ∈ L ∪ L0 then one can verify that a vector x satisfies M x ≥ b, l ≤ x ≤ u, if and only if the vector x0 defined by x0i = −x0i0 = xi for all i ∈ V satisfies a0i x0i + a0j x0j ≥ b0ij for all ij ∈ E 0 , x0i ≥ li and x0i0 ≥ −ui for all i ∈ V . Since the inequalities x0i ≥ −x0i0 , i ∈ V , are valid for the latter system, as they are the inequalities a0i x0i + a0i0 x0i0 ≥ b0ii0 for the edges ii0 of G0 , then the polyhedron defined by M x ≥ b, l ≤ x ≤ u corresponds to the face of the polyhedron defined by a0i x0i + a0j x0j ≥ b0ij , ij ∈ E 0 , x0i ≥ li , x0i0 ≥ −ui , i ∈ V given by x0i = −x0i0 , i ∈ V . Given an I ∪ I 0 -path P in G0 , this determines an inequality as in (3.10) for the instance given by G0 , b0 and I ∪ I 0 . Substituting x0i for −x0i0 , for every i ∈ V , into such inequality, we obtain the inequality of (3.3) relative to the I-trail T obtained from P by identifying the pairs of nodes i, i0 for every i ∈ V such that i, i0 are in P . Since, by Corollary 3.8, the system obtained from a0i x0i + a0j x0j ≥ b0ij , ij ∈ E 0 , x0i ≥ li , x0i0 ≥ −ui , i ∈ V by juxtaposing the inequalities of the form (3.10) relative to I ∪ I 0 -paths of G0 is totally dual integral, and since, by Theorem 1.17, setting to equality some inequalities of a system preserves total dual integrality, then the system obtained from the above by setting x0i = −x0i0 , i ∈ V , is totally dual integral, therefore also the system defined by (3.2) and (3.3) for every I-trail P is totally dual integral. This concludes the proof of the claim. ¦ We conclude the proof of Theorem 3.1. Given a vector c ∈ ZV , we show how to get an integral optimal solution for the dual of min{c> x : x satisfies (3.2), (3.3) for every I-path P } from an integral optimal solution y for the dual of min{c> x : x satisfies (3.2), (3.3) for every I-trail P }. In fact, if T is an I-trail that is not an I-path such that yT > 0, then there exists a cycle C, a node j and two trails Q, R such that T = Q, j, C, j, R. Note that S = Q, j, R is an I-trail with the same endnodes of T but with

44

Bipartite vertex covers with parity conditions

less cycles than T . Since the edge-node incidence matrix of G is totally unimodular, by Theorem 1.10 (iv), the edges of C can be partitioned in two subsets C 1 and C 2 such that any two adjacent edges of C are contained in the same subset if and only if one of them has a −1 and the other has a +1 in their common endnode. Moreover, we may assume that C 1 has cost at least db(C)/2e. One can verify that, by our choice of the partition C 1 , C 2 , the integral vector y 0 that is identical to y except for yS0 = yS + yT , ye0 = ye + yT , ∀e ∈ C 1 , yT0 = 0, is feasible for the dual of min{c> x : x satisfies (3.2), (3.3) for every I-trail P }. Furthermore its objective value is at least that of y plus yT (db(S)/2e + db(C)/2e − d(b(S) + b(C))/2e) ≥ 0, thus y 0 is also optimal. Since the total number of cycles contained in I-trails whose associated dual variables are positive strictly decreases, by repeating the argument we obtain an integral optimal solution for the dual of min{c> x : x satisfies (3.2), (3.3) for every I-path P }. ¤

3.4 3.4.1

Polynomial time solvability Bipartite case

The proof of Theorem 3.4 gives an algorithm (albeit not a polynomial time one) to derive an integral optimal solution y ∗ for (3.6) from an integral optimal solution y˜ for (3.8), as follows. Initially we set y ∗ := 0. Each time we apply a reduction, we update the value of y ∗ and then apply our algorithm recursively on the reduced problem as long as the current vector c is not the all zero vector. Since each time we apply a reduction the value P of some entry of c decreases, the total number of iterations is bounded by i∈U ∪W ci , which is not a polynomial bound on the size of the problem. ˜ contains a pair of symmetric loaded edges, then for More in detail: if G each e ∈ E we update ye∗ := ye∗ + min{˜ ye1 , y˜e2 }, where e1 and e2 are the ˜ images of e in G, apply the reduction on the symmetric edges, and proceed recursively on the reduced problem. ˜ has a directed cycle, we can find in polynomial time a directed cycle If D C that either is symmetric or has at most one node in I. If C is symmetric, then it has non-positive cost, thus we apply the reduction on C, update yR∗ := yR∗ + γ, where R is the I-path defined by the edges with images in C and γ is the minimum value of y˜ on the loaded edges of C, and proceed recursively on the reduced problem. Otherwise, we augment on the cycle among C and sym(C) with cost zero by the smallest load on its edges, and apply the reduction on the symmetric edges.

3.4 Polynomial time solvability

45

˜ is acyclic and there exists a non-isolated node in I, then we can find If D ˜ in polynomial time a symmetric directed path of non-positive cost S in D starting from some node of in-degree 0, we apply the reduction on S, update yR∗ := yR∗ + γ, where R is the I-path defined by the edges with images in S and γ is defined as in the proof, and proceed recursively on the reduced problem. If all nodes of I are isolated, we can find in polynomial time a directed path S of cost zero from a node of in-degree zero to a node of out-degree zero. We augment on S by the minimum among c˜j , for every endnode j of S incident with an unloaded arc of S, and min{˜ ye : e ∈ S, y˜e > 0}, and we apply the reduction on the symmetric edges. Notice that each iteration can be performed in strongly polynomial time. While we cannot give a polynomial bound on the number of iterations of the algorithm described above, we can prove that the number of iterations in which we apply a reduction on a symmetric path or symmetric cycle of non-positive cost is bounded by the number of edges of G. In fact, each time we apply a reduction on a symmetric cycle, the number of loaded edges decreases by at least one. We apply the reduction on a symmetric path of non-positive cost S only when S starts in a node with in-degree 0 and ends in a node with out-degree 0. In this case, if k is the only endnode of S incident with an unloaded arc of S, we have X y˜e = c˜k = c˜sym(k) ≥ min{˜ ye : e ∈ S, y˜e > 0} c˜k − e3k

since k is incident only with unloaded arcs, and sym(k) is incident with a loaded arc of S. Thus, each time we apply a reduction on a symmetric path, the number of loaded edges decreases by at least one. So, if on a given instance the algorithm described above does not perform any reduction on the symmetric edges, then it performs at most |E| iterations. In particular, this happens if and only if the optimal solution y ∗ for (3.6) produced by the algorithm satisfies ye∗ = 0 for every e ∈ E. We will show next that we can reduce to this case, thus proving the following. Theorem 3.9. There is a strongly polynomial-time algorithm to compute an integral optimal solution for (3.6) for each integral vector c for which it has a finite optimum. Proof. Let x∗ be an integral optimal solution for the problem min{c> x : x satisfies (3.5)} and let x˜ be the solution for (3.7) defined by x˜i = x∗i , i ∈ I, x∗ x∗ x˜i = b 2i c, x˜i0 = d 2i e, i ∈ L. By Theorem 3.4, x∗ is optimal if and only if x˜ is optimal for (3.7), and c> x∗ = c˜> x˜. Notice that this remains true even if c

46

Bipartite vertex covers with parity conditions

is not an integral vector. Given e = ij ∈ E, let αe ∈ RU ∪W be the coefficient vector of the constraint of (3.4) relative to e, that is αie = ai , αje = aj , αke = 0 for k ∈ (U ∪ W ) \ {i, j}. Claim. Given e¯ ∈ E such that αe¯x∗ = be¯, one can compute in strongly polynomial time the maximum γ such that x∗ remains optimal for the problem min{(c − γαe¯)> x : x satisfies (3.5)}. ˜ I : x˜i > 0}, F = {ij ∈ E˜ : x˜i +˜ Proof of claim. Let J = {i ∈ U˜I ∪ W xj > bij }. ˜I ∪W ˜I U By complementary slackness, a vector y˜ ∈ R is optimal for (3.8) if and only if y˜ satisfies P y˜ = c˜i Pe3i e ˜e ≤ c˜i e3i y y˜e = 0 y˜e ≥ 0

i∈J ˜)\J i ∈ (U˜ ∪ W e∈F e ∈ E˜ \ F.

(3.11)

˜ Let µ = max{s : s ≤ y˜e1 , s ≤ Let e1 , e2 ∈ E˜ be the images of e¯ in E. y˜e2 , y˜ satisfies (3.11)}. We show that γ = µ. ˜ ˜ We first show γ ≤ µ. Let c0 = c−γαe¯, and c˜0 ∈ RUI ∪WI be the correspond˜ Since x∗ is optimal for min{(c − γαe¯)> x : ing cost vector on the nodes of G. x satisfies (3.5)}, then x˜ is optimal for the problem (3.7) with respect to the cost vector c˜0 . Hence there exists a vector y˜0 that satisfies P y˜0 = c˜0i Pe3i e0 ˜e ≤ c˜0i e3i y 0 y˜e = 0 y˜e0 ≥ 0

i∈J ˜)\J i ∈ (U˜ ∪ W e∈F e ∈ E˜ \ F.

Now the vector defined by y˜e = y˜e0 + γ if e ∈ {e1 , e2 }, y˜e = y˜e0 otherwise, satisfies (3.11) and γ ≤ y˜e1 , γ ≤ y˜e2 . ˜ ˜ Now we show that γ ≥ µ. Let c0 = c − µαe¯, and c˜0 ∈ RUI ∪WI be the ˜ If y˜ is the solution that satisfies corresponding cost vector on the nodes of G. (3.11) and maximizes s, then the vector defined by y˜e0 = y˜e − µ if e ∈ {e1 , e2 }, y˜e0 = y˜e otherwise, satisfies (3.11) with respect to the cost vector c˜0 . Hence x˜ is optimal for the problem (3.7) with respect to the cost vector c˜0 , and x∗ is optimal for the problem min{(c − µαe¯)> x : x satisfies (3.5)}. Finally, since the coefficients of the variables in (3.11) and in s ≤ y˜e1 , s ≤ y˜e2 , are 0, ±1, γ can be computed in strongly polynomial time using an algorithm of Tardos [33] (see Corollary 1.6). ¦

3.4 Polynomial time solvability

47

Let e1 , . . . , em be the edges in E such that αe x∗ = be , e ∈ {e1 , . . . , em }. k Set c0 = c and, for k = 1, . . . , m, let ck = ck−1 −bγk cαe , where γk is the maxk imum γ such that x∗ remains optimal for the problem max{(ck−1 − γαe )> x : x satisfies (3.5)}. By the previous claim, we can compute c1 , . . . , cm in strongly polynomial time. Given any integral optimal solution y ∗ for the dual of min{cm> x : x satisfies (3.5)}, then the vector y¯, defined by y¯ek = ye∗k + bγk c for every k = 1, . . . , m, y¯P = yP∗ for every I-path P , is an integral optimal solution for (3.6). By definition of γ1 , . . . , γm , c1 , . . . , cm , for every e ∈ E we must have ye∗ < 1, thus ye∗ = 0. This concludes our proof, since we have shown above that in this case the algorithm given by the proof of Theorem 3.4 finds an integral optimal solution for min{cm> x : x satisfies (3.5)} in strongly polynomial time. This completes the proof of the claim.

3.4.2

General case

Theorem 3.10. There is a strongly polynomial-time algorithm to compute an integral optimal solution for the dual of min{c> x : x satisfies (3.2), (3.3) for every I-path P } whenever the problem has a finite optimum. Proof. We showed in Theorem 3.9 that an integral optimal solution for the dual of any problem of the form min{c> x : x satisfies (3.5)} can be computed in strongly polynomial time for each integral vector c for which it has a finite optimum. The proof of Corollary 3.8 shows how to obtain, in strongly polynomial time, an integral optimal solution for the dual of any problem of the form min{c> x : x satisfies (3.10)} from an integral optimal solution for the dual of a problem of the form min{c> x : x satisfies (3.5)} with integer requirements bij − ai li − aj lj . The proof of the Claim in the proof Theorem 3.1 shows how to reduce, in strongly polynomial time, any problem of the form min{c> x : x satisfies (3.2), (3.3) for every I-trail P } to a problem of the form min{¯ c> x¯ : 0 x¯ satisfies (3.10)} in some auxiliary graph G , but with a polynomial number of inequalities (of the form x0i + x0i0 ≥ 0) set to equality. The next claim shows that an integral optimal solution of the dual of any problem in the latter form can be computed in strongly polynomial time. Finally, in the last part of the proof of Theorem 3.1, we showed how to get an integral optimal solution for the dual of min{c> x : x satisfies (3.2), (3.3) for every I-path P } from an integral optimal solution for the dual of min{c> x : x satisfies (3.2), (3.3) for every I-trail P }. Notice that the procedure described terminates in strongly polynomial time.

48

Bipartite vertex covers with parity conditions

Claim. Let Ax ≥ b, Cx ≥ d be a totally dual integral system of linear inequalities, where A ∈ Zp×n and C ∈ Zq×n . Let α = max{kAk∞ , kCk∞ }. Given c ∈ Zn , let γ be the q-dimensional vector with all entries equal to n!αn kck∞ and c¯ = c + C > γ. If (y ∗ , u∗ ) is an integral optimal solution for the dual of min{¯ c> x | Ax ≥ b, Cx ≥ d}, (3.12) (where y and u are relative to the rows of A and C, respectively) then (y ∗ , u∗ − γ) is an integral optimal solution for the dual of min{c> x | Ax ≥ b, Cx = d},

(3.13)

provided that the latter has a finite optimum. Proof of claim. Clearly (y ∗ , u∗ − γ) is integral and feasible for the dual of (3.13). We show it is optimal. Let x¯ be an optimal solution of (3.13), and (¯ y , u¯) be an optimal basic solution for the dual of (3.13). Since (¯ y , u¯) is basic, then the absolute values of its components are bounded above by kck∞ times the maximum among the absolute values of the determinants of the square submatrices of (A> , C > ), which is at most αn n!. Therefore u¯ ≥ −γ. Thus (¯ y , u¯ + γ) is feasible for the dual of (3.12), x¯ is feasible for (3.12), and > c¯x¯ = b y¯ + d> u¯ + γ > C x¯ = b> y¯ + d> (¯ u + γ), thus x¯ and (¯ y , u¯ + γ) are optimal for (3.12) and its dual, respectively. Thus b> y¯ + β u¯ = b> y ∗ + β(u∗ − γ), so (y ∗ , u∗ − γ) is an integral optimal solution for the dual of (3.13). This concludes the proof of the claim. ¦ In particular, if the system Ax ≥ b, Cx ≥ d is of the form (3.10), and the number of rows of C is bounded by some polynomial in n, then, for any c ∈ Zn , the problem of finding an integral dual solution of (3.13) can be reduced in strongly polynomial time to the problem of finding an integral dual solution of (3.12), which by Theorem 3.9 can be solved in strongly polynomial time.

Chapter 4 A class of matrices arising from bidirected graphs In this chapter we study totally half-modular matrices obtained from {0, ±1}matrices with at most two nonzero entries per column by multiplying by 2 some of the columns. We show that the matrix   1 1 2 0 M4 =  −1 0 0 2  0 1 0 2 is the only minor minimal matrix in such class (up to multiplying rows and columns by −1) that does not have the Edmonds-Johnson property. Notice that this result extends Theorem 2.2 of Edmonds and Johnson. We will also show that, for each matrix M in this class that does not contain M4 as a minor, one can minimize in polynomial time any linear function over the integer hull of b ≤ M x ≤ c, l ≤ x ≤ u, for all integral vectors b, c, l, u. The results in this chapter are joint work with A. Musitelli and G. Zambelli [10].

4.1

Introduction

Given a bidirected graph G and a subset F of its edges, we denote by A(G, F ) the matrix obtained from Σ(G) by multiplying by 2 the columns corresponding to the edges in F . We will show in Lemma 4.5 that a matrix A(G, F ) is totally half-modular if and only if (G, F ) satisfies the following. Cycles condition: no odd cycle of G contains edges in F .

50

A class of matrices arising from bidirected graphs

By Theorem 2.1 and Theorem 2.9, the class of matrices A(G, F ) with the Edmonds-Johnson property, and such that (G, F ) satisfies the cycles condition is closed under taking minors. In this chapter we characterize the pairs (G, F ) that satisfy the cycles condition for which A(G, F ) has the Edmonds-Johnson property. Our result implies that Conjecture 2.1 is true if A is a totally half-modular matrix with at most two nonzero entries per column and such that all the nonzero elements in a column have the same absolute value. In fact, the only minor that we need to exclude is A3 , because A4 never appears. Let G = (V, E, σ) be a bidirected graph and F ⊆ E. Given a node v ∈ V , 0 the signing σ 0 obtained from σ by setting σv,e = −σv,e for all edges e incident with v is said to be obtained by switching signs on the node v. 0 Given e = vw ∈ E, the signing σ 0 obtained from σ by setting σv,e = −σv,e , 0 σw,e = −σw,e , is said to be obtained by switching signs on the edge e. Given a node v ∈ V , the pair (G0 , F 0 ) obtained from (G, F ) by deleting node v is defined as follows. V (G0 ) = V \ {v}, E(G0 ) contains all edges of E not incident with v and a loop on w for each edge vw ∈ E with v 6= w. We will identify such loops in G0 with the corresponding edges incident with v in G. The signing on the edges of G0 coincides with σ on G \ v, while F 0 = F ∩ E(G0 ). Notice that our definition of node deletion is non-standard, since we do not remove all the edges incident with v, but we replace them with loops. (See Figure 4.1 for an example.) Given a subset of nodes U ⊆ V , the pair (G0 , F 0 ) is obtained from (G, F ) by deleting the nodes in U if (G0 , F 0 ) is obtained from (G, F ) by deleting one by one the nodes in U .

(G,F)

(G’,F’)

-

-

v -

-

Figure 4.1: Node deletion: (G0 , F 0 ) is obtained from (G, F ) by deleting the node v: the boldfaced edges represent the edges in F and in F 0 , and the signing on the edges is everywhere +1 except where a − occurs. Given an edge e ∈ E, (G0 , F 0 ) is obtained from (G, F ) by deleting edge

4.1 Introduction

51

e if G0 = (V, E \ {e}, σ 0 ) and F 0 = F \ {e}, where σ 0 coincides with σ on E \ {e}. Let e = vw ∈ E such that e is not a loop and σv,e 6= σw,e . We say that 0 (G , F 0 ) is obtained from (G, F ) by contracting edge e if G0 is the bidirected graph obtained by replacing the nodes v, w with one new node r ∈ / V , by deleting all the edges vw with σv,vw 6= σw,vw , by replacing each edge vw with σv,vw = σw,vw with a loop in r with sign σv,vw , which we identify with the original edge in E, by replacing each edge tv, t 6= w or tw, t 6= v, with an edge tr in E(G0 ), which we identify with the original edge in E, and by letting the signing in G0 coincide with σ on E(G0 ). Let F 0 be the union of F and the set of the loops in r corresponding to edges vw in G with the same sign in their endnodes. Notice that, if e = vw ∈ E \ F , then A(G0 , F 0 ) is obtained by pivoting the entry (v, e) in A(G, F ), and by removing the row corresponding to v and the column corresponding to e. Moreover, if e = vw ∈ F and v is incident only with edges in F , then A(G0 , F 0 ) is obtained by dividing by 2 the row corresponding to v, then by pivoting the entry (v, e) in A(G, F ), and by removing the row corresponding to v and the column corresponding to e. Furthermore, if (G, F ) contains an odd cycle C, then the pair (G0 , F 0 ) obtained from (G, F ) by contracting one by one the edges in E(C), contains a new loop l ∈ F 0 in the node obtained from the contraction of C. Given a pair (G, F ), we call a pair (G0 , F 0 ) a minor of (G, F ) if it is obtained by the latter through some of the following operations: (O1) switching signs on a node or on an edge of G; (O2) deleting a node or an edge in (G, F ); (O3) contracting an edge in E(G) \ (F ∪ L(G)); (O4) contracting an edge vw in F \ L(G) such that δ(v) ⊆ F . By the above discussion, the class of pairs (G, F ) that satisfy the cycles condition and such that A(G, F ) has the Edmonds-Johnson property is closed under taking minors. One can verify that if the pair (G, F ), which we name G4 , is as in figure 4.2, then A(G, F ) does not have the Edmonds-Johnson property. The following is the main result of this chapter. Theorem 4.1. Given a pair (G, F ) that satisfies the cycles condition, A(G, F ) has the Edmonds-Johnson property if and only if (G, F ) does not contain G4 as a minor.

52

A class of matrices arising from bidirected graphs

v2 e1 e3 v1 e4 e2 v3 Figure 4.2: G4 : the boldfaced edges represent the edges in F , and the signing σ of G is everywhere +1 except for σv2 ,e1 = −1. Notice that the following partial result was shown by Del Pia and Zambelli [12]. Theorem 4.2. Given a pair (G, F ) that satisfies the cycles condition and that does not contain G3 as a minor, then A(G, F ) has the Edmonds-Johnson property. Where the pair (G, F ), which we name G3 , is as in figure 4.3.

v2 e1 v1

e3 e2 v3

Figure 4.3: G3 : the boldfaced edges represent the edges in F , and the signing σ of G is everywhere +1 except for σv2 ,e1 = −1. Notice that, in terms of bidirected graphs, Theorem 2.2 states that, given a bidirected graph G and a subset F of its loops, A(G, F ) has the EdmondsJohnson property. Since for every bidirected graph G and for every subset F of its loops, (G, F ) satisfies the cycles condition and does not contain G4 as a minor, Theorem 4.1 reduces to Theorem 2.2 in this case. Notice that Theorem 4.1 implies the following Corollary 4.3. Given a totally half-modular matrix A with at most two nonzero entries per column, such that all the nonzero elements in a column

4.2 First Chv´ atal closure

53

have the same absolute value, then A has the Edmonds-Johnson property if and only if it does not contain A3 as a minor. Proof. If A contains A3 as a minor, then A does not have the EdmondsJohnson property, because by Observation 2.11, A3 does not have the Edmonds-Johnson property. Let (G, F ) be a pair such that A = A(G, F ). If A does not contain A3 as a minor, then (G, F ) does not contain G4 as a minor, since A(G4 ) contains A3 as a minor (pivot on the +1 entry corresponding to the node v1 and the edge e1 ), and by Theorem 4.1, A has the EdmondsJohnson property.

In what follows let C be the family of pairs (G, F ), where G is a bidirected graph and F is a subset of its edges, such that (G, F ) satisfies the cycles condition and does not contain G4 as a minor. Hence the matrices A(G, F ) with (G, F ) in C define another class of totally half-modular matrices with the Edmonds-Johnson property. On the algorithmic side, we will show that, if A(G, F ) is a matrix in our class, one can minimize in polynomial time any linear function over the integer hull of b ≤ A(G, F )x ≤ c, l ≤ x ≤ u, for all integral vectors b, c, l, u. In contrast, Conforti et Al. [6] proved that deciding if a system of the form A(G, F )x ≥ b has an integral solution is N P-complete even if Σ(G) is totally unimodular. We recall that, by Theorem 1.13, Σ(G) is totally unimodular if and only if G does not contain any odd cycle. Therefore, if Σ(G) is totally unimodular, the cycles condition is always verified. In the next section we show how we can reduce ourselves to study systems of the form A(G, F )x = c, x ≥ 0; we describe the irredundant Chv´atal inequalities for such systems and for the more general systems A(G, F )x = c, 0 ≤ x ≤ u. Moreover we show how to separate such Chv´atal inequalities in polynomial time. In Section 4.4 we finally prove Theorem 4.1.

4.2

First Chv´ atal closure

Let G = (V, E, σ) be a bidirected graph and F ⊆ E. By definition A(G, F ) has the Edmonds-Johnson property if the system b ≤ A(G, F ) x ≤ c l≤x≤u

(4.1)

54

A class of matrices arising from bidirected graphs

has Chv´atal rank at most 1 for every b, c ∈ ZV , l, u ∈ ZE . Next we show that, in proving Theorem 4.1, we can reduce ourselves to study systems of the form A(G, F ) x = c (4.2) x ≥ 0, where c ∈ ZV . Lemma 4.4. If (4.2) has Chv´atal rank at most 1 for every (G, F ) in C and every integral c, then A(G, F ) has the Edmonds-Johnson property for every (G, F ) in C . Proof. At first we show that if (4.2) has Chv´atal rank at most 1 for every (G, F ) in C and every integral c, then the system A(G, F )x = c 0≤x≤u

(4.3)

has Chv´atal rank at most 1 for every (G, F ) in C and every integral c, u. Let (G, F ) be a pair in C , let c, u be integral vectors and let x¯ be a vector in the first closure of (4.3). Now let (G0 , F 0 ), c0 and x¯0 be obtained from (G, F ), c and x¯ in the following way. For each nonloop edge e = v1 v2 in E \ F (resp. in F ), add a new node ve and replace e with the path v1 , v1 ve , ve , ve v2 , v2 such that the edges v1 ve and ve v2 are in E(G0 ) \ F 0 (resp. in F 0 ), such that v1 ve and ve v2 have a +1 sign in the vertex ve , the edge v1 ve has in v1 the same sign that e had in v1 , while the edge ve v2 has in v2 the opposite sign that e had in v2 , decrease c0v2 by σv2 ,e ue (resp. by 2σv2 ,e ue ), set c0ve = ue (resp. c0ve = 2ue ), and set x¯0v1 ve = x¯e and x¯0ve v2 = ue − x¯e . Similarly, for each loop e in E \ F (resp. in F ) incident with a node v, add a new node ve and replace e with the edge vve and with a loop le in ve such that the edges vve and le are in E(G0 ) \ F 0 (resp. in F 0 ), such that vve and le have a +1 sign in the vertex ve , the edge vve has in v the same sign that e had in v, set c0ve = ue (resp. c0ve = 2ue ), and set x¯0vve = x¯e and x¯0le = ue − x¯e . Notice that also (G0 , F 0 ) is in C . By Lemma 1.24 (i), x¯0 is in the first closure of A(G0 , F 0 )x0 = c0 , x0 ≥ 0. If the latter system has Ch´atal rank at most 1, then x¯0 is a convex combination of integral solutions. Hence x¯ can be expressed as a convex combination of integral vectors satisfying (4.3). Now we show that the system b ≤ A(G, F )x ≤ c 0≤x≤u

(4.4)

4.2 First Chv´ atal closure

55

has Chv´atal rank at most 1 for every (G, F ) in C and every integral b, c, u. Let (G, F ) be a pair in C , let b, c, u be integral vectors and let x¯ be a vector in the first closure of (4.4). Now let G0 be obtained from G by adding a loop ev ∈ / E(G) on every node v of G. Notice that also (G0 , F ) is in C . 0 Clearly A(G0 , F ) = (A I). Let u0 ∈ ZE(G ) be obtained from u by setting u0ev = cv − bv for every new loop µ ev . ¶ x¯ 0 By Lemma 1.24 (i), x¯ = is in the first closure of the c − A(G, F )¯ x system A(G0 , F )x0 = c, 0 ≤ x0 ≤ u0 . If the latter system has Ch´atal rank at most 1, then x¯0 is a convex combination of integral solutions. Hence x¯ is a convex combination of integral vectors satisfying (4.4). Finally we show that A(G, F ) has the Edmonds-Johnson property for every (G, F ) in C . Let b, c, l, u be integral vectors and let x¯ be a vector in the first closure of (4.2). Let b0 = b − Al, c0 = c − Al, u0 = u − l, x¯0 = x¯ − l. It is easy to see that x¯0 is in the first closure of b0 ≤ A(G, F )x0 ≤ c0 , 0 ≤ x0 ≤ u0 . If the latter system has Ch´atal rank P at most 1, then x¯0 is a P convex combination i 0 of integral solutions, i.e. x¯ = i∈I λi y˜ , 0 ≤ λi ≤ 1, i∈I λi = 1. Hence P x¯ = i∈I λi y i , where y i = y˜i + l is a convex combination of integral vectors satisfying (4.1). To describe the Chv´atal inequalities for our systems, we need total halfmodularity. Thus at first we prove the following lemma. Lemma 4.5. A(G, F ) is totally half-modular if and only if (G, F ) satisfies the cycles condition. Proof. Assume (G, F ) satisfies the cycles condition, and let A0 be a nonsingular square submatrix of A(G, F ). Let v1 , . . . , vk be the nodes of G that correspond to the rows of A0 and let e1 , . . . , ek be the edges of G that correspond to the columns of A0 . Let G0 be the bidirected graph whose nodes are v1 , . . . , vk and whose edges are e1 , . . . , ek , where an edge ei that has only one endnode among v1 , . . . , vk is a loop of G0 . Then A0 is obtained from the sign matrix Σ(G0 ) of G0 by multiplying by 2 some of its columns. If G0 is not connected, then A0 is a block diagonal matrix and so we can look at each block. Hence, by symmetry, we may assume that e1 , . . . , ek−1 induce a spanning tree of G0 . If ek is a loop of G0 , then Σ(G0 ) is totally unimodular, thus A0−1 is half-integral because it is obtained from Σ(G0 )−1 by dividing some rows by 2. Thus ek has two distinct endnodes and it is contained in the unique cycle C of G0 . One can readily verify that, if C is even, then Σ(G0 )

56

A class of matrices arising from bidirected graphs

is singular, and so is A0 . Therefore C µis odd. ¶ Up to permuting rows and P Q columns, we may assume that Σ(G0 ) = , where P is the incidence 0 R matrix of the cycle C. It is easy to check that det(P ) = ±2, while R is −1 −1 totally unimodular, therefore µ P −1 is half-integral ¶while R is integral. One −1 −1 P −P QR can verify that Σ(G0 )−1 = . Notice that the first |C| 0 R−1 rows of Σ(G0 )−1 are half-integral, while the other rows are integral. The matrix A0−1 is obtained from Σ(G0 )−1 by dividing by 2 the rows corresponding to the edges in F . Since (G, F ) satisfies the cycles condition, C ∩ F = ∅, therefore A0−1 is obtained from Σ(G0 )−1 by dividing by 2 some of the last k − |C| rows. Hence A0−1 is half-integral. Viceversa, if there exists an odd cycle C of G such that C ∩ F 6= ∅, then A(C, F ∩ C)−1 is obtained by dividing by 2 some of the rows of Σ(C)−1 . Since C is odd, all the nonzero entries of Σ−1 (C) have value ±1/2, hence A(C, F ∩ C)−1 is not half-integral. At some point in our proof of Theorem 4.1 it will be convenient to introduce some upper bounds on the system (4.2). Hence in the following Lemma we describe the Chv´atal inequalities for these more general systems. Lemma 4.6. Let (G, F ) be a pair satisfying the cycles condition. Let αx ≥ β be an irredundant nontrivial Chv´atal inequality for A(G, F ) x = c 0 ≤ x ≤ u.

(4.5)

Then there exists a connected set U ⊆ V (G), and a partition E 0 , E u of the edges in δ(U ) \ F , such that c(U ) + u(E u ) is odd and αx ≥ β is equivalent to x(E 0 ) − x(E u ) ≥ −u(E u ) + 1.

(4.6)

Furthermore, there is no nontrivial partition U1 , U2 of U such that all the edges between U1 and U2 are in F . Proof. Let A = A(G, F ). By Lemma 2.8 and Lemma 2.10, αx ≥ β is equivalent to an inequality of the form (µA + γ 0 − γ u )x ≥ dµc − γ u ue, where µ ∈ {0, ± 21 }V (G) , γ 0 , γ u ∈ {0, 12 }E(G) , µA + γ 0 − γ u is integral, and µc − γ u u is not integral. Let U be the set of nodes corresponding to non-zero entries of µ. Notice that all entries of µA are integer, except for the entries corresponding to edges in δ(U ) \ F , which have value ± 21 . Hence there exists a partition E u , E 0 of δ(U ) \ F such that γe0 = 21 if e ∈ E 0 , 0 otherwise, and γeu = 21 if e ∈ E u , 0 otherwise. Since c(U ) + u(E 0 ) is odd, then

4.2 First Chv´ atal closure

57

dµc − γ u ue = µc − γ u u + 12 . Since µAx = µc for every x that satisfies (4.5), αx ≥ β is equivalent to (γ 0 − γ u )x ≥ −γ u u + 12 . Multiplying the latter by 2, one obtains (4.6). Finally, if there is a nontrivial partition U1 , U2 of U such that all the edges between U1 and U2 are in F , let Eh0 = E 0 ∩ δ(Uh ), Ehu = E u ∩ δ(Uh ), h = 1, 2. Notice that δ(Uh ) \ F = Eh0 ∪ Ehu and that (δ(U1 ) \ F ) ∩ (δ(U2 ) \ F ) = ∅. Also, since c(U ) + u(E u ) is odd, by symmetry we may assume c(U1 ) + u(E1u ) is odd and c(U2 ) + u(E2u ) is even. Hence x(E10 ) − x(E1u ) ≥ −u(E1u ) + 1 is a Chv´atal inequality, while x(E20 ) − x(E2u ) ≥ −u(E2u ) is implied by (4.2). The sum of the two latter inequalities is precisely (4.6), contradicting the assumption that αx ≥ β is irredundant. We will refer to inequalities of the form (4.6) as odd-cut inequalities (relative to U , E 0 , E u ). Notice that, when G is an undirected simple graph, F = ∅, c is the vector of all 1s, while u is the vector with all entries equal to +∞, the odd-cut inequalities reduce to the well known ones for the perfect matching polytope. 0 u Remark 4.7. An odd-cut inequalityP relative to U, P E , E is satisfied by x¯ ∈ E(G) R satisfying (4.5) if and only if e∈E 0 x¯e + e∈E u (ue − x¯e ) ≥ 1.

Lemma 4.8. Let (G, F ) be a pair satisfying the cycles condition, and A = A(G, F ). The first closure of {x : Ax = c, 0 ≤ x ≤ u} is the intersection of the first closure of {x : Ax = c, 0 ≤ xe ≤ ue , ∀e ∈ E(G) \ F } and the set {x : 0 ≤ xf ≤ uf , ∀f ∈ F }. Proof. The statement follows from Lemma 4.6, and the fact that the oddcut inequalities for {x : Ax = c, 0 ≤ x ≤ u} are precisely the odd-cut inequalities for {x : Ax = c, 0 ≤ xe ≤ ue , ∀e ∈ E(G) \ F }. Most of the times, in the proof of Theorem 4.1, we will be considering systems in the form (4.2). Hence in the following Lemma we describe the Chv´atal inequalities for these simpler systems. Lemma 4.9. Let (G, F ) be a pair satisfying the cycles condition. Let αx ≥ β be an irredundant nontrivial Chv´atal inequality for (4.2). Then there exists a connected set U ⊆ V (G), such that c(U ) is odd and αx ≥ β is equivalent to x(δ(U ) \ F ) ≥ 1. (4.7) Furthermore, there is no nontrivial partition U1 , U2 of U such that all the edges between U1 and U2 are in F .

58

A class of matrices arising from bidirected graphs

Proof. Directly from Lemma 4.6, by taking u = +∞. The following lemma will be useful in the proof of Theorem 4.1. Lemma 4.10. Let G be a bidirected graph, let F ⊆ E(G), and let I ⊆ F . If the system A(G, F )x = c, x ≥ 0 has Chv´atal rank at most 1 for every integral c, then the system A(G, F )x = c, x ≥ 0, xf ≤ 1, f ∈ I has Chv´atal rank at most 1 for every integral c. Proof. By contradiction assume that x¯ is a fractional vertex of the first closure of A(G, F )x = c, x ≥ 0, xf ≤ 1, f ∈ I. Let x¯0e = x¯P / I, e for every e ∈ 0 0 xf c. x¯f = x¯f − b¯ xf c for every f ∈ I. Moreover let cv = cv − 2 f ∈δ(v)∩I σv,f b¯ Clearly x¯0 satisfies A(G, F )x = c0 , x ≥ 0. Notice that c0v ≡2 cv for every v ∈ V (G). Hence for every U ⊆ V (G), c0 (U ) ≡2 c(U ). Thus the odd-cut inequalities for A(G, F )x = c0 , x ≥ 0 and for A(G, F )x = c, x ≥ 0, xf ≤ 1, f ∈ I are the same, and since x¯0e = x¯e for every e ∈ E(G) \ F , x¯0e is a vertex of the first closure of A(G, F )x = c0 , x ≥ 0. A contradiction, as it is fractional.

4.2.1

Algorithmic aspects

We consider the separation problem for the odd-cut inequalities. Consider a pair (G, F ), where G = (V, E, σ) is a bidirected graph and F ⊆ E. Let c ∈ ZV , and x¯ ∈ RE be a vector satisfying A(G, F )x = c, x ≥ 0. Clearly the odd-cut inequalities relative to such system are valid for its first closure. We want to determine whether there exists an odd-cut inequality violated by x ¯. Let G0 = (V, E 0 , σ 0 ) be the bidirected graph obtained from G by deleting all edges in F , and by appending a loop fv on v ∈ V for each edge f ∈ F 0 incident with v, with sign σv,f = σv,f . We let F 0 be the set of such loops, v 0 0 and σv,e = σv,e for each e ∈ E \ F 0 . 0 Let x¯0 ∈ RE be obtained from x¯ by removing the components relative to edges in F and by adding a component for each loop fv ∈ F 0 with value x¯0fv = x¯f . Notice that x¯0 satisfies A(G0 , F 0 ) x0 = c x0 ≥ 0,

(4.8)

and the odd-cut inequalities for (4.8) are precisely the odd-cut inequalities for (4.2) since, by definition, the odd-cut inequalities are independent on the edges in F . Thus x¯0 violates an odd-cut inequality for (4.8) if and only if x¯ violates the same odd-cut inequality for (4.2). Thus we only need to determine if there exists an odd-cut inequality for (4.8) violated by x¯0 . Notice

4.3 Balanced bipartitions

59

that, by construction, the sum of the absolute values of the entries of each column of A(G0 , F 0 ) is at most 2, hence by Theorem 2.2, A(G0 , F 0 ) has the Edmonds-Johnson property, even when A(G, F ) does not. Therefore the oddcut inequalities are sufficient to define the convex hull of integral solutions of (4.8). Edmonds and Johnson [16] show that the problem of finding an integral solution of (4.8) minimizing a given linear function can be solved in polynomial time, therefore also the separation problem over the convex hull of integral solutions of (4.8) can be solved in polynomial time with the ellipsoid method [23]. This shows the following. Theorem 4.11. There is a polynomial time algorithm that, given x¯ ∈ RE satisfying A(G, F )x = c, x ≥ 0, returns either an odd-cut inequality violated by x¯, or determines that none exists. Corollary 4.12. Let (G, F ) satisfy the cycles condition. There is a polynomial time algorithm that, for any α ∈ RE , finds a vector x∗ in the first Chv´ atal closure of A(G, F )x = c, x ≥ 0 minimizing αx. Proof. By Lemma 4.9, the only nontrivial inequalities valid for the first Chv´atal closure of A(G, F )x = c, x ≥ 0 are the odd-cut inequalities. Since these can be separated in polynomial time, with the ellipsoid method [23] we can solve the minimization problem in polynomial time. Corollary 4.13. Let (G, F ) be in C . There is a polynomial time algorithm that, for any α ∈ RE , finds a integral vector x∗ satisfying A(G, F )x = c, x ≥ 0 minimizing αx. Proof. Follows immediately from Theorem 4.1 and Corollary 4.12.

4.3

Balanced bipartitions

Let G be a bidirected graph and F ⊆ E(G). Given two disjoint sets R, B of the edges in E(G), we say that R, B is a balanced bipartition of the edges in R ∪ B, if for every v ∈ V (G), 1 2

X vw∈E(G)\F vw∈R

σv,vw +

X vw∈F vw∈R

σv,vw =

1 2

X vw∈E(G)\F vw∈B

σv,vw +

X

σv,vw .

(4.9)

vw∈F vw∈B

In the proof of Theorem 4.1, it will be useful at times to determine balanced bipartitions. Remark 4.14. If there exists a balanced bipartition of the edges of E(G), then

60

A class of matrices arising from bidirected graphs

a) |δG (v) \ F | is even for every v ∈ V ; ¯ of G \ F such that L(G) ¯ = ∅, |δG (V (G))| ¯ b) for every component G is ¯ congruent modulo 2 to the number of odd edges in E(G). Proof. Condition a) follows immediately from the fact that, given a balanced bipartition R, B, ¶ µ X X 1 σv,vw − σv,vw 2 vw∈E(G)\F vw∈R

vw∈E(G)\F vw∈B

is an integer. ¯ be a connected component of G \ F such To prove condition b), let G ¯ ¯ we obtain that L(G) = ∅). Summing equations 4.9 for all v ∈ V (G), X ¯ vw∈R∩E(G)

−

X

(σv,vw + σw,vw ) − 2

−

X

¯ vw∈B∩E(G)

X

(σv,vw + σw,vw ) +

vw∈F ∩R\L(G) ¯ v,w∈V (G)

X

(σv,vw + σw,vw )+

vw∈F ∩B\L(G) ¯ v,w∈V (G)

¯ vw∈δG (V (G))∩R ¯ v∈V (G)

σv,vw +

(σv,vw + σw,vw ) = 2

X

σv,vw .

¯ vw∈δG (V (G))∩B ¯ v∈V (G)

¯ is congruent modulo 2 to |δG (V (G))|, ¯ Hence the number of odd edges in E(G) ¯ since σv,vw + σw,vw ≡2 0 for every vw ∈ F \ L(G) with v, w ∈ V (G). Let G be a bidirected graph and F ⊆ E(G) such that (G, F ) ∈ C and (G, F ) satisfies conditions a) and b) of Remark 4.14. Suppose that G has two distinct blocks B1 , B2 such that, for i = 1, 2, either Bi contains an odd cycle or E(Bi ) ∩ F 6= ∅. Let w be a cutnode of G separating B1 and B2 . Choose V1 , V2 ⊂ V such that V1 ∪ V2 = V (G), V1 ∩ V2 = {w}, V (B1 ) ⊆ V1 , V (B2 ) ⊆ V2 , the graphs induced by V1 and V2 are connected, there is no edge between V1 \ {w} and V2 \ {w}. Partition the edges of E(G) into two sets E1 , E2 so that, for every e ∈ Ei , all endnodes of e are in Vi , i = 1, 2 (notice that such a partition is not uniquely defined, since a loop on w can be put arbitrarily in E1 or E2 ). Let Gi = (Vi , Ei ) and Fi = Ei ∩ F , i = 1, 2. Notice that, by condition a) of Remark 4.14, then either |δG1 (w) \ F1 | and ˜ 1 , F˜1 ) and |δG2 (w) \ F2 | are both odd, or they are both even. We define (G ˜ 2 , F˜2 ) as follows: (G

4.3 Balanced bipartitions

61

˜ i the graph obtained 1. If |δG1 (w)\F1 | and |δG2 (w)\F2 | are odd, we define G from Gi by adding a loop `i on w with sign +1, and F˜i = Fi , i = 1, 2. 2. If |δG1 (w) \ F1 | and |δG2 (w) \ F2 | are both even, and one among (G1 , F1 ) and (G2 , F2 ) does not satisfy condition b) of Remark 4.14, we define ˜ i the graph obtained from Gi by adding a loop `i on w with sign +1, G and F˜i = Fi ∪ {`i }, i = 1, 2. 3. If both (G1 , F1 ) and (G2 , F2 ) satisfy conditions a) and b) of Remark 4.14, ˜ i , F˜i ) = (Gi , Fi ), i = 1, 2. let (G ˜ 1 , F˜1 ) and (G ˜ 2 , F˜2 ) are obtained from (G, F ) by breaking We say that (G B1 and B2 at w. Lemma 4.15. Let G be a bidirected graph and F ⊆ E(G) such that (G, F ) ∈ ˜ 1 , F˜1 ) and C and (G, F ) satisfies conditions a) and b) of Remark 4.14. If (G ˜ 2 , F˜2 ) are obtained from (G, F ) ∈ C by breaking B1 and B2 at w, then (G ˜ 1 , F˜1 ) and (G ˜ 2 , F˜2 ) are in C , and they satisfy conditions a) and b) of (G Remark 4.14. ˜ 1 , F˜1 ) and (G ˜ 2 , F˜2 ) have a balanced bipartition, then Furthermore, if both (G also (G, F ) has a balanced bipartition. ˜ 1 , F˜1 ) and (G ˜ 2 , F˜2 ) satisfy conditions a) and b) Proof. We first show that (G ˜ 1 , F˜1 ) and (G ˜ 2 , F˜2 ) satisfy of Remark 4.14. It is immediate to see that (G condition a), and they trivially satisfy condition b) if case 1) or 3) applies. ¯ 1 be a Assume case 2) applies, and that (G1 , F1 ) violates condition b). Let G ¯ 1 ) = ∅ and the number of odd connected component of G1 \ F such that L(G ¯ ¯ 1 ))|. Notice that any edges of E(G1 ) is not congruent modulo 2 to |δG1 (V (G connected component of G1 \ F1 or G2 \ F2 that does not contain node w is ¯ 1 ), so the number of also a connected component of G \ F . Hence w ∈ V (G ¯ ¯ ˜ 1 , F˜1 ) odd edges of E(G1 ) is congruent modulo 2 to |δG˜ 1 (V (G1 ))|, therefore (G ˜ 2 , F˜2 ) violates condition b), then the connected satisfies condition b). If (G ¯ 2 of G2 \ F that contains w satisfies L(G ¯ 2 ) = ∅ and the number component G ¯ 2 ) is not congruent modulo 2 to |δ ˜ (V (G ¯ 2 ))|, therefore of odd edges of E(G G2 ¯ ¯ 2 ))|. the number of odd edges of E(G2 ) is congruent modulo 2 to |δG2 (V (G ¯ of G \ F containing w violates condition Thus the connected component G b). ˜ i , F˜i ) ∈ C , i = 1, 2. This is obvious if case 1) or 3) above We show that (G apply, since (Gi , Fi ) is a minor of (G, F ), so (Gi , Fi ) ∈ C . Since, in case 1), ˜ i , F˜i ) is obtained from (Gi , Fi ) by adding a loop `i not in Fi , then also (G ˜ i , F˜i ) ∈ C. If case 2) applied, by symmetry it is sufficient to show that (G ˜ 1 , F˜1 ) is a minor of (G, F ). Indeed, consider the pair (G0 , F 0 ) obtained (G

62

A class of matrices arising from bidirected graphs

from (G, F ) by contracting all the edges in E2 \ F . Notice that F 0 \ E1 is nonempty, since either E(B2 ) ∩ F 6= ∅, or B2 contains an odd cycle C, and when we contract all edges in C we obtain a new loop in F 0 . Thus there exists an edge f ∈ F 0 \ E1 with one endnode in w, say f = vw. Deleting from (G0 , F 0 ) all edges in F 0 \ E1 except f , and deleting all nodes in V (G0 ) \ V1 , ˜ 1 , F˜1 ). we obtain (G ˜ 1 , F˜1 ) and (G ˜ 2 , F˜2 ) have a balanced biWe finally show that, if both (G partition, then also (G, F ) has a balanced bipartition. Let Bi , Ri be a bal˜ i , F˜i ), for i = 1, 2. If case 3) above applies, then anced bipartition of (G B1 ∪ B2 , R1 ∪ R2 is a balanced bipartition of (G, F ). If cases 1) or 2) apply, assuming that li ∈ Bi , i = 1, 2, then (B1 ∪ R2 ) \ {l1 }, (R1 ∪ B2 ) \ {l2 } is a balanced bipartition of (G, F ). Next we give a few technical lemmas that will be used in the proof of Theorem 4.1 to show that certain pairs (G, F ) ∈ C have balanced bipartitions. Lemma 4.16. Let (G, F ) be a pair in C such that G\F is connected. Assume that there exists a family T of trails such that: (C1) each edge in E(G) \ (F \ L(G)) is contained in exactly one trail in T and no edge in F \ L(G) is contained in any trail in T ; (C2) if a trail T in T contains edges in L(G) \ F , then it contains exactly two of them and they are its first and last edge. If a trail T in T does not contain edges in L(G) \ F , then it is closed and |E(T ) ∩ F | ≡2 |{vw ∈ E(T ) \ F : σv,vw = σw,vw }|; (C3) for every f ∈ F \ L(G), there exists a trail in T that contains both endnodes of f . Then there exists a balanced bipartition of the edges in E(G). Proof. We give a construction for the bipartition. At first we give a balanced bipartition of the edges in E(G) \ (F \ L(G)), and later we use it to obtain a balanced bipartition of the edges in E(G). We give a bipartition of the edges of E(G) \ (F \ L(G)) in the following way: For every trail T = v1 , e1 , . . . , ek−1 , vk in T , ej and ej−1 are in the same subset if and only if σvj ,ej 6= σvj ,ej−1 , for every j = 2, . . . , k − 1. Notice that this is a balanced bipartition of the edges in E(G) \ (F \ L(G)). For every edge f ∈ F \ L(G), we define a subtrail T (f ) of a trail in T recursively in the following way. Among all the edges f ∈ F \ L(G) for which

4.3 Balanced bipartitions

63

we have not yet defined T (f ), let g = vw be the one such that there exists a subtrail of a trail in T , from v to w, of minimal length. Let T (g) be such minimal length subtrail. Notice that, by construction, for every edge f ∈ F \ L(G), T (f ) is a subtrail of a trail in T , the endnodes of T (f ) are the two endnodes of f , and all the interior nodes of T (f ) are different from the endnodes of f . Now we show that, for every pair f, g of edges in F \ L(G), either T (f ) is a subtrail of T (g), or T (g) is a subtrail of T (f ), or E(T (f )) ∩ E(T (g)) = ∅. By contradiction, let f, g ∈ F \ L(G), f 6= g be such that E(T (f )) ∩ E(T (g)) 6= ∅, T (f ) is not a subtrail of T (g), and T (g) is not a subtrail of T (f ). Therefore, an internal node of T (f ) is an endnode of g, and an internal node of T (g) is an endnode of f . By symmetry we can assume that T (f ) was defined before T (g), thus the length of T (f ) is less than or equal to the length of T (g). It follows that it is not possible that both the endnodes of g appear in T (f ), otherwise we would have defined T (g) before T (f ). Thus, by considering the closed trail T (f ), f , and by deleting the endnode of g that does not appear in T (f ), we get G4 as a minor, a contradiction. Now notice that, for every f = vw ∈ F \ L(G), all the cycles contained in the closed trail f, T (f ) are even. Otherwise suppose that at least one cycle C contained in the closed trail f, T (f ) is not even. Notice that C cannot contain f , since (G, F ) satisfies the cycles condition. Hence by contracting all the edges in C we get a new loop l in F . By using a cycle in f, T (f ) containing f , and a path in T (f ) from such cycle to l we get G4 as a minor. For the same reason, for every f = vw in F \ L(G), T (f ) does not contain loops in F ∩ L(G). Therefore, by construction, the edge ev in T (f ) incident with v and P the edge ew in T (f ) incident with w are in the same subset if and only if e∈E(T (f )) σr,e ≡4 0. Since all the cycles contained in f, T (f ) are r∈e,r6=v,w

even, and since in T (f ) there are no loops in F , ev and ew are in the same subset if and only if σv,ev + σv,f + σw,f + σw,ew ≡4 0. We say that an edge f in F \L(G) is maximal in F \L(G) if for every other edge g in F \L(G), either T (g) is a subtrail of T (f ), or E(T (f ))∩E(T (g)) = ∅. Now we extend our bipartition of E(G) \ (F \ L(G)) to a bipartition of E(G) by doing the following recursively for every f maximal among the edges in F \ L(G) not in our bipartition. Let v be an endnode of f and let e be the only edge in T (f ) incident with v. Put f in the same side of the bipartition of e if and only if σv,f = σv,e , and switch the current bipartition of every edge in T (f ). By construction, the bipartition defined above is balanced. Remark 4.17. Let (G, F ) ∈ C such that G \ F is connected, and let ∆ ⊆ F ∩L(G). Assume that there exists a family T of trails that satisfy conditions

64

A class of matrices arising from bidirected graphs

(C1)-(C3) and such that ∆ ⊆ T¯ for some T¯ ∈ T . Let T = l1 , T1 , l2 , . . . , Tk−1 , lk be the subtrail of T¯ such that ∆ = {l1 , . . . , lk }. Furthermore, assume that ∆ = E(T ) ∩ F , and that (V (T ), E(T )) is bipartite. Then the balanced bipartition given in the proof of Lemma 4.16 has the property that, for every i =P1, . . . , k − 1, liP and li+1 are in the same side of the partition if and only if e∈{li ,E(Ti ),li+1 } v∈e σv,e ≡4 0. Lemma 4.18. Let G be a bipartite bidirected graph and let F ⊆ E(G) such that G \ F is connected. If (G, F ) satisfies the following (i) F is a star; (ii) |δG (v) \ F | is even for every v ∈ V (G); (iii) if L(G) ⊆ F , then |L(G)| is even; then there exists a balanced bipartition of E(G). Proof. Notice that, since F is a star and G is bipartite, then (G, F ) does not contain the G4 minor. Thus (G, F ) ∈ C . If L(G) ⊆ F , then let T be an Eulerian circuit on the edges of E(G) \ (F \ L(G)). We want to prove that the family T = {T } satisfies conditions (C1)-(C3) of Lemma 4.16. It is immediate to see that (C1) and (C3) hold. We show that (C2) holds. By hypothesis (iii) in the statement, |L(G)| is even, thus we need to show that |{vw ∈ E(G) \ F : σv,vw = σw,vw }| is even. Since |δG (v) \ F | is even for every v ∈ V (G), then E(G) \ F can be partitioned into cycles, and each cycle in G is even, because G is bipartite, thus |{vw ∈ E(G) \ F : σv,vw = σw,vw }| is even. Hence, by Lemma 4.16 there exists a balanced bipartition R, B of the edges in E(G). Otherwise |L(G) ⊆ F | ≥ 1, and since |δG (v) \ F | is even for every v ∈ V (G), |L(G) \ F | is even, hence |L(G) \ F | ≥ 2. Let l1 and l2 be two loops in L(G)\F . By Theorem 1.13, since G is bipartite, V (G) can be partitioned into two sets V 1 and V 2 so that, for every edge e = vw ∈ E(G) \ L(G), v, w are ˜ obtained in distinct sets if and only if σv,e = σw,e . We define a new graph G from G by adding a new vertex r and by replacing every loop lv in L(G) \ F incident with a node v and different from l1 and l2 , with an edge vr with σv,vr = σv,lv , and with sign σr,vr = −σv,lv if v ∈ V 1 , σr,vr = σv,lv if v ∈ V 2 . ˜ \ L(G), ˜ v, w Notice now that, by construction, for every edge e = vw ∈ E(G) 1 2 are not both in V ∪ {r} or not both in V if and only if σv,e = σw,e . By

4.4 Proof of Theorem 4.1

65

˜ is bipartite. As before, since G ˜ is bipartite and F is a star, Theorem 1.13, G ˜ F) ∈ C. (G, ˜ with its corresponding edge Notice that, if we identify every edge of G ˜ defines a balanced bipartition in G, a balanced bipartition of the edges of G ˜ F ) has a balanced of the edges in G, hence we only need to show that (G, ˜ ˜ bipartition. Now let T be an Eulerian circuit on the edges of E(G)\(F \L(G)) such that its first edge is l1 and its last edge is l2 . Notice that T = {T } ˜ F, L(G). ˜ satisfies conditions (C1)-(C3) of Lemma 4.16 with respect to G, Hence, by Lemma 4.16 there exists a balanced bipartition of the edges in ˜ E(G).

Lemma 4.19. Let (G, F ) ∈ C such that G is bipartite and G\F is connected. Assume (G, F ) satisfies conditions a) and b) of Remark 4.14 and that there exists a path P in G \ F that passes through all the nodes of G incident with some edge in F . Then there exists a partition of the edges in E(G) \ F in one closed trail, if L(G) ⊆ F , or, if L(G) \ F 6= ∅, in |L(G) \ F |/2 trails such that their first and last edge are in L(G) \ F , such that one of the trails passes through all the nodes incident with edges in F . Proof. Let P be a path in E(G)\F that passes through all the nodes incident with edges in F , and let v and w be its endnodes. Let G0 be obtained from G by removing the edges in E(P ) and by adding one loop lv in v, and one loop lw in w. Notice that G0 \F may be not connected. Now partition the edges in E(G0 ) \ F into |L(G0 ) \ F |/2 trails starting and ending in loops in E(G0 ) and in closed trails. If the loops lv and lw are contained in different trails, say T1 and T2 , then we get from T1 , P, T2 one new trail starting and ending in loops in E(G) \ F different from the artificial lv , lw . Otherwise the loops lv and lw are contained in the same trail. Hence there exists a partition of the edges in E(G) \ F in |L(G) \ F |/2 trails starting and ending in loops in E(G) \ F and in closed trails, where one of these closed trail passes through all the nodes incident with edges in F . In both cases, since G \ F is connected, we can always get the partition of the edges in E(G) \ F required by recursively combining together the closed trails with another trail of the partition.

4.4

Proof of Theorem 4.1

Claim 4.1. A(G4 ) does not have the Edmonds-Johnson property. Proof of claim. Notice that the matrix A3 can be obtained from A(G4 ) by pivoting on the +1 entry corresponding to the node v1 and the edge e1 , and

66

A class of matrices arising from bidirected graphs

then by removing the column corresponding to the edge e1 . Since A3 does not have the Edmonds-Johnson property, then by Theorem 2.1, A(G4 ) does not have the Edmonds-Johnson property. ¦ In the remainder of the section we will prove that for any (G, F ) in C , the system (4.2) has Ch´atal rank at most 1 for any integral vector c. By Lemma 4.4, this will imply Theorem 4.1. By contradiction, suppose that there exists a pair (G, F ) in C and an integral vector c such that the first closure of system (4.2) has a fractional vertex x¯. Among all such counterexamples, P choose (G, F ), c, x¯ such that the quadruple (|V (G)|, |E(G) \ L(G)|, |E(G)|, b e∈E(G) x¯e c) is lexicographically minimal. Let A = A(G, F ). Given a node v, if G0 is obtained from G by switching sign on node v and c0 ∈ RV (G) is defined by c0u = cu , u ∈ V (G) \ {v}, c0v = −cv , then it is immediate to verify that x¯ is a vertex of the first closure of A(G0 , F )x = c0 , x ≥ 0, because for every U ⊆ V (G), c(U ) is odd if and only if c0 (U ) is odd. So, if (G, F ), c, and x¯ are a minimal counterexample, then also (G0 , F ), c0 , and x¯ is a minimal counterexample. Hence, throughout the proof we will perform such switching whenever convenient. Notice that (G, F ) has at least an edge in F \ L(G), otherwise, by Theorem 2.2, A has the Edmonds-Johnson property, contradicting our choice of (G, F ). Furthermore, G is connected. If not, let G0 be a component of G such that x¯e ∈ / Z for some e ∈ E(G0 ), let F 0 = F ∩ E(G0 ), and let A0 = A(G0 , F 0 ). Let x¯0 and c0 be the restrictions of x¯ and c, respectively, to E(G0 ). Notice that (G0 , F 0 ) is in C and that |V (G0 )| < |V (G)|, hence the system A0 x0 = c0 , x0 ≥ 0 has Chv´atal rank at most 1. Then clearly x¯0 is a fractional vertex of the first closure of A0 x0 = c0 , x0 ≥ 0, a contradiction. We say that an inequality αx ≤ β is tight at x¯ if α¯ x = β. Notice that, since x¯ is a vertex of the first closure of (4.2), it satisfies at equality |E(G)| linearly independent inequalities among the ones in (4.2) and the odd-cut inequalities. Claim 4.2. x¯e > 0 for every e ∈ E(G). Proof of claim. If not, assume that there exists an edge e in E(G) with x¯e = 0. Let (G0 , F 0 ) be obtained from (G, F ) by deleting e, and let A0 = A(G0 , F 0 ). Let x¯0 be the vector obtained from x¯ by removing the component corresponding to e. Since (G0 , F 0 ) is in C , |V (G0 )| = |V (G)|, |E(G0 )\L(G0 )| ≤ |E(G)\L(G)|, and |E(G0 )| < |E(G)|, the system A0 x0 = c, x0 ≥ 0 has Chv´atal rank at most 1. Since the vector c has not changed, the odd-cut inequalities

4.4 Proof of Theorem 4.1

67

for A0 x0 = c, x0 ≥ 0 are exactly the odd-cut inequalities for (4.2). Moreover notice that x¯0 (δ(U ) \ F 0 ) = x¯(δ(U ) \ F ) for every U ⊆ V (G). Hence x¯0 is a fractional vertex of the first closure of A0 x0 = c, x0 ≥ 0, a contradiction. ¦ Let F = {U ⊆ V | x¯(δ(U ) \ F ) = 1}. By Claim 4.2, x¯ is the unique solution of the system A(G, F )x = c x(δ(U ) \ F ) = 1

U ∈ F.

By Lemma 2.4, we can choose a laminar subfamily L of F such that x¯ is the unique solution of the system A(G, F )x = c x(δ(U ) \ F ) = 1

U ∈ L,

and the elements of {χ(δ(U ) \ F ) : U ∈ L } are not linear combination of rows of A(G, F ). In particular, |E| ≤ |V | + |L |. Claim 4.3. For every e ∈ E(G), 0 < x¯e < 1. Furthermore, for every e ∈ E(G) \ F there exists U ∈ L such that e ∈ δ(U ). Proof of claim. By Claim 4.2, x¯e > 0 for every e in E(G). We show that x¯f < 1 for any f in F . Let f be an edge in F , and suppose x¯f ≥ 1. Notice that, possibly by switching the signs on the endnode/s of f , we can assume that f has a sign +1 on its endnode/s. Let x¯0 be obtained from x¯ by decreasing by 1 the component corresponding to f and let c0 be obtained from c by decreasing by 2 the P component/s corresponding to the endnode/s P of f . Since b e∈E(G) x¯0e c < b e∈E(G) x¯e c, by minimality of (G, F ), c, x¯ the system Ax = c0 , x ≥ 0 has Chv´atal rank at most 1. Notice that each entry of c0 has the same parity as the corresponding entry of c, therefore the odd-cut inequalities for Ax = c0 , x ≥ 0 are exactly the odd-cut inequalities for (4.2). Since such inequalities do not depend on the values on the edges in F , then x¯0 is a fractional vertex of the first closure of the system Ax = c0 , x ≥ 0, a contradiction. We show next that for every e in E(G) \ F there exists U ∈ L such that e ∈ δ(U ). By contradiction, suppose there exists e ∈ E(G) \ F such that for every U ∈ L, e ∈ / δ(U ). If e = vw is not a loop, possibly by switching signs on

68

A class of matrices arising from bidirected graphs

v we may assume that σv,e 6= σw,e . Let (G0 , F 0 ) be obtained from (G, F ) by contracting e, let r be the node obtained from the contraction of vw, and let A0 = A(G0 , F ). Let x¯0 be the vector obtained from x¯ by removing the components corresponding to the removed edges, and let c0 be obtained from c by removing the components corresponding to v and w and adding a component relative to r with value c0r = cv + cw . Since (G0 , F 0 ) is in C and |V (G0 )| < |V (G)|, the system A0 x0 = c0 , x0 ≥ 0 has Chv´atal rank at most 1. Notice that x¯0 satisfies the system A0 x0 = c0 , x0 ≥ 0, and that the equation (A0 x0 )r = c0r is the sum of (Ax)v = cv and (Ax)w = cw . Furthermore, the odd-cut inequalities for A0 x0 = c0 , x0 ≥ 0 are exactly the odd-cut inequalities for (4.2) relative to sets U ⊆ V (G) that either contain both v and w or none of them. Notice that all the odd-cut inequalities corresponding to sets in L are of this form, since e ∈ / δ(U ) for every U ∈ L , hence x¯0 is in the first closure of A0 x0 = c0 , x0 ≥ 0 and, since x¯e > 0, it satisfies at equality |E(G)|−1 ≥ |E(G0 )| linearly independent inequalities among A0 x0 = c0 , x0 ≥ 0 and the odd-cut inequalities corresponding to sets in L . Therefore x¯0 is a vertex of the first closure of A0 x0 = c0 , x0 ≥ 0, so it is integral. Hence x¯e must be the only fractional entry in x¯, which is impossible since (A¯ x)v = cv which is integer. If e is a loop on node v, notice that, in this case, the column relative to e in the constraint matrix of the system Ax = c, x(δ(U ) \ F ) ≥ 1, U ∈ L , is the vector of all zeroes except in row Av . Since the columns of said matrix are linearly independent, then e is the only loop of G on v. Let (G0 , F 0 ) be obtained from (G, F ) by deleting node v and let A0 = A(G0 , F 0 ). Clearly 0 (G0 , F 0 ) is in C . Let x¯0 ∈ ZE(G ) be the vector obtained from x¯ by removing 0 the components corresponding to the deleted loops on v, and let c0 ∈ ZV (G ) be obtained from c by removing the component corresponding to v. Since |V (G0 )| < |V (G)|, A0 x0 = c0 , x0 ≥ 0 has Chv´atal rank at most 1. Notice that A0 is obtained from A by removing the row corresponding to v and the column relative to e. Thus x¯0 is in the first closure of A0 x0 = c0 , x0 ≥ 0, since the odd-cut inequalities for the latter system are the odd-cut inequalities for (4.2) relative to sets U ⊆ V (G) \ {v}. Since U ⊆ V (G) \ {v} for every U ∈ L , all odd-cut inequalities for (4.2) corresponding to sets in L are valid for the first closure of A0 x0 = c0 , x0 ≥ 0. Since x¯e > 0, x¯0 satisfies at equality |E(G)| − 1 = |E(G0 )| linearly independent inequalities among A0 x0 = c0 and the odd-cut inequalities relative to sets U ∈ L . Hence x¯0 is a vertex of the first closure of A0 x0 = c0 , x0 ≥ 0, therefore x¯0 is integral. Thus x¯e is the only fractional entry of x¯, which is impossible since (A¯ x)v = cv which is integer. Therefore, given e in E(G) \ F , x¯e ≤ 1 since there exists U ∈ L such that e ∈ δ(U ), and since x¯(δ(U )) = 1. We show that, given e in E(G) \ F , x¯e < 1. By contradiction, suppose that there exists an edge e in E(G) \ F

4.4 Proof of Theorem 4.1

69

such that x¯e = 1. Possibly by switching signs on the endnode/s of e, we may assume that e has sign +1 on its endnode/s. Let U ⊆ V (G) define an odd-cut inequality in L such that e ∈ δ(U ). Hence e is the only edge in δ(U ) \ F . Let (G0 , F 0 ) be obtained from (G, F ) by deleting e, and let A0 = A(G0 , F 0 ). Let c0 be obtained from c by subtracting 1 to the entries relative to the endnode/s of e, and let x¯0 be the vector obtained from x¯ by removing the component corresponding to e. Since (G0 , F 0 ) is in C , |V (G0 )| = |V (G)|, |E(G0 ) \ L(G0 )| ≤ |E(G) \ L(G)|, and |E(G0 )| < |E(G)|, the system A0 x0 = c0 , x0 ≥ 0 has Chv´atal rank at most 1. We show that x¯0 is in the first closure of A0 x0 = c0 , x0 ≥ 0. Clearly x¯0 satisfies the system, so we need to show that it satisfies the odd-cut inequalities. Let S ⊆ V (G0 ) such that c0 (S) is odd. By Lemma 4.9 we may assume that S cannot be partitioned into two nonempty sets S1 , S2 such that the only edges in G0 between S1 and S2 are in F . Therefore, since δG0 (U ) ⊆ F , either S ⊆ U or S ⊆ V (G) \ U . If no endnode of e is in S, then clearly x¯0 (δG0 (S) \ F ) ≥ 1. Hence we assume e ∈ δG (S), hence c(S) is even. If S ⊆ U , then c(U \ S) is odd, hence x¯0 (δG0 (S) \ F ) = x¯(δG (U \ S) \ F ) ≥ 1. If S ⊆ V (G) \ U , then c(U ∪ S) is odd, hence x¯0 (δG0 (S) \ F ) = x¯(δG (U ∪ S) \ F ) ≥ 1. Therefore x¯0 is in the first closure of A0 x0 = c0 , x0 ≥ 0, hence it is a convex combination of integral vectors y 1 , . µ . . , yk ¶ satisfying such system. Thus x¯ = µ ¶ x¯e x¯e is a convex combination of , i = 1, . . . , k, which are integral x¯0 yi vectors satisfying (4.2), a contradiction. ¦ Claim 4.4. G does not contain a cycle in F . Proof of claim. By contradiction, suppose C is a cycle in F . Since (G, F ) ∈ C , the cycle C is even and so there is an even number of nodes of C incident with two edges of C with the same sign on that node. Hence the edges of C can be partitioned in two subsets R and B such that any two adjacent edges of C are contained in the same subset if and only if they have distinct signs on their common endnode. Now let y = x¯ + ²χ(R) − ²χ(B) and let z = x¯ − ²χ(R) + ²χ(B). By Claim 4.3, there exists ² > 0 such that both y and z satisfy all the inequalities of (4.2). Moreover, both y and z satisfy all the odd-cut inequalities for (4.2), since they only involve variables relative to edges in E(G) \ F . Since x¯ = 12 (y + z), x¯ is not a vertex of the first closure of (4.2), a contradiction. ¦ Claim 4.5. Each node in V (G) is incident with at least one edge in E(G)\F . Proof of claim. By contradiction let v be a node in V (G) incident only with edges in F . Notice that we can assume that G contains at least two nodes,

70

A class of matrices arising from bidirected graphs

otherwise it is clear that A(G, F ) has the Edmonds-Johnson property. Thus, since G is connected, there exists an edge in F \ L(G) incident with v. Thus let f = vw be an edge in F \ L(G). Possibly by switching sign on v, we may assume that σv,f 6= σw,f . Notice that cv is even, as otherwise the odd-cut corresponding to the set {v} is not satisfied. Let (G0 , F 0 ) be obtained from (G, F ) by contracting f , let r be the node obtained from the contraction of v and w, and let A0 = A(G0 , F 0 ). Clearly, (G0 , F 0 ) is in C . Let x¯0 , be the vector obtained from x¯ by removing the component corresponding to f , and let c0 be obtained from c by removing the components corresponding to v and w and adding a new component relative to r with value c0r = cv + cw . Since |V (G0 )| < |V (G)|, the system A0 x0 = c0 , x0 ≥ 0 has Chv´atal rank at most 1. Since cv is even, c0r has the same parity as cw . Thus, given U ⊆ V (G) \ {v}, then c(U ) is odd if and only if c0 (U 0 ) is odd, where U 0 = U if w ∈ / U , U 0 = U \ {w} ∪ {r} if w ∈ U . Moreover, notice that δ(U ) \ F = δG0 (U 0 ) \ F , therefore the odd-cut inequalities for A0 x0 = c0 , x0 ≥ 0 are exactly the odd-cut inequalities for (4.2) relative to U with U ⊆ V (G)\{v}. Notice that these inequalities are precisely the odd-cut inequalities for (4.2). Clearly x¯0 satisfies also the system A0 x0 = c0 , x0 ≥ 0, so it is in its first closure. Since the inequality (A0 x0 )r = c0r is the sum of (Ax)w = cw and (Ax)v = cv , and since all the odd-cut inequalities for (4.2) are also valid for A0 x0 = c0 , x0 ≥ 0, there are |E(G)| − 1 = |E(G0 )| linearly independent inequalities valid for the first closure of A0 x0 = c0 , x0 ≥ 0 tight at x¯0 , hence x¯0 is a vertex of such first closure, and it is therefore integral. So x¯e = x¯0e is integer for every e in E(G0 ), contradicting Claim 4.3. ¦

4.4.1

Half-integrality for some special cases

Claim 4.6. If G \ F is connected and V (G) ∈ / L , then x¯ is half-integral. Proof of claim. Let U be a maximal set in the laminar family L . Notice that, since L is laminar, for every S ∈ L either S ⊆ U or S ⊆ V (G) \ U . Since V (G) ∈ / L , then U ⊂ V (G). Since G \ F is connected, there exists at least one edge e ∈ E(G) \ (F ∪ L(G)) such that e ∈ δ(U ). Let e = vw and let v be the endnode of e in U . Now let (G0 , F ) be obtained from (G, F ) by deleting e and adding loops ev , ew on v and w with signs σv,e and σw,e , respectively. Let A0 = A(G0 , F ). One can readily verify that (G0 , F ) is in the class C , |V (G0 )| = |V (G)|, and |E(G0 ) \ L(G0 )| < |E(G) \ L(G)|, thus A0 x0 = c, x0 ≥ 0 has Chv´atal rank at most 1. Now let x¯0 be obtained from x¯ by replacing the component corresponding to the edge

4.4 Proof of Theorem 4.1

71

e with two components equal to the replaced one and that correspond to the new loops ev and ew . Clearly x¯0 satisfies the system A0 x0 = c, x0 ≥ 0. Each odd-cut inequality of the latter system is satisfied by x¯0 since, for every S ⊆ V , x¯0 (δG0 (S) \ F ) ≥ x¯(δG (S) \ F ), where equality holds if and only if |S ∩ {v, w}| ≤ 1. Thus x¯0 is in the first closure of A0 x0 = c, x0 ≥ 0. Since U is a maximal set in L , |S ∩ {v, w}| ≤ 1 for every S ∈ L , hence x¯0 (δG0 (S) \ F ) = 1 for every S ∈ L .

(4.10)

Therefore, since x¯ satisfies tightly |E(G)| inequalities of the first closure of (4.2), x¯0 satisfies tightly |E(G)| = |E(G0 )| − 1 linearly independent inequalities of the first closure of A0 x0 = c, x0 ≥ 0. Hence x¯0 lies on a face of dimension 1 of the first closure of A0 x0 = c, x0 ≥ 0. This implies that x¯0 is a convex combination of two vertices y and z of the first closure of such system, i.e. x¯0 = λy + (1 − λ)z, 0 ≤ λ ≤ 1. Since A0 x0 = c, x0 ≥ 0 has Chv´atal rank at most 1, then y and z are integral, so 0 < λ < 1. By (4.10), y(δG0 (S) \ F ) = z(δG0 (S) \ F ) = 1 for every S ∈ L . By Claim 4.3, each edge h ∈ E(G) \ F is in δ(S) for some set S ∈ L , thus each edge h ∈ E(G0 )\(F ∪{ew }) is in δ(S) for some set S ∈ L . Therefore yh , zh ∈ {0, 1} for every h ∈ E(G0 ) \ (F ∪ {ew }). Thus, since x¯0ev = x¯0ew = x¯e < 1, we can assume that yev = 1 and zev = 0 and that precisely one among yew and zew is 0. Hence x¯e = λ. If zew = 0, then yew = 1 since x¯0ew = λyew , thus if we define two points y¯, z¯ ∈ RE(G) by y¯h = yh , h ∈ E(G) \ {e}, y¯e = 1, and z¯h = zh , h ∈ E(G) \ {e}, z¯e = 0, then y¯ and z¯ are integral points satisfying (4.2) and x¯ = λ¯ y + (1 − λ)¯ z , contradicting the fact that x¯ is a vertex of the first closure of (4.2). Therefore yew = 0 and zew = k for some positive integer k. Since x¯e = λyev + (1 − λ)zev = λyew + (1 − λ)zew , then λ = k/(k+1). If k = 1, then all components of x¯ are equal to 1/2 and we are done. Thus we may assume that k ≥ 2. This implies x¯e = k/(k+1) > 1/2. Moreover, w ∈ / S for every S ∈ L , otherwise ew ∈ δG0 (S) and z(δG0 (S) \ F ) = 1 would imply zew = 1. Notice also that since z(δG0 (U ) \ F ) = 1 there exists an edge g ∈ δG0 (U ) \ F such that zg = 1. Hence δG (U ) \ F = {e, g} and x¯g = 1 − λ = 1/(k + 1) < 1/2. If g is not a loop, then we may apply to g the same argument as above, thus obtaining that x¯g > 1/2, a contradiction. Thus g is a loop. Next we show that x¯0 is in the first closure of the system A0 x0 = c, x0 ≥ 0, xew ≤ 1. By Lemma 4.6, the irredundant non trivial inequalities for the first closure of A0 x0 = c, x0 ≥ 0, xew ≤ 1 are valid for the first closure of A0 x0 = c, x0 ≥ 0, or they are of the form x0 (δG0 (S)\(F ∪{ew }))−x0ew ≥ 0 for S ⊆ V (G) such that c(S) is even, w ∈ S, and S cannot be partitioned into two nonempty sets S1 and S2 such that all the edges between S1 and S2 are in F . Since all

72

A class of matrices arising from bidirected graphs

edges of E(G0 ) between U and V (G) \ U are in F , then the latter inequalities are defined by sets S contained in V (G)\U such that w ∈ S and c(S) is even. We only need to show that x¯0 satisfies such inequalities. Given S ⊆ V (G) \ U with w ∈ S and c(S) even, c(U ∪ S) is odd, hence x¯(δG (U ∪ S) \ F ) ≥ 1. Notice that δG0 (S) \ (F ∪ {ew }) = δG (U ∪ S) \ (F ∪ {g}), hence x¯0 (δG0 (S) \ (F ∪ {ew })) − x¯0ew = x¯(δG (U ∪ S) \ F ) − x¯e − x¯g k 1 ≥ 1− − = 0. k+1 k+1 Hence x¯0 lies on the face of dimension 1 of the first closure of A0 x¯0 = c, x ≥ 0, x0ew ≤ 1 defined by A0 x0 = c and x0 (δG0 (S) \ F ) = 1, S ∈ L . Notice that y is a vertex of such face. Let z 0 be the other vertex. Notice that z 0 is strictly inside the segment between y and z, i.e. z 0 = µy + (1 − µ)z for some 0 < µ < 1, so ze0 w > 0 and z 0 is not integral. Therefore z 0 is not a vertex of the first closure of A0 x0 = c, x0 ≥ 0. Thus there exists a set W ⊂ V (G) \ U such that w ∈ W , c(W ) is even, x¯0 (δG0 (W ) \ (F ∪ {ew })) − x¯0ew = 0, and the |E(G0 )| inequalities 0

A0 x0 = c x0 (δG0 (S) \ F ) ≥ 1, x0 (δG0 (W ) \ (F ∪ {ew })) − x0ew ≥ 0

S∈L

are linearly independent. Notice that, since w ∈ / S for every S ∈ L , then all of the above inequalities are valid also for the first closure of A0 x0 = 0, x0h ≥ 0 e ∈ E(G0 ) \ {ew }, x0ew ≤ 1. A0 x0 = 0, x0h ≥ 0, x0ew ≤ 1

h ∈ E(G0 ) \ {ew },

Let G00 be the bidirected graph G00 = (V (G0 ), E(G0 ), σ 0 ), where σh0 = σh for every h ∈ E(G0 ) \ {ew }, and σe0 w = −σew , and let A00 = A(G00 , F ). Notice that A00 is obtained from A0 by multiplying by −1 the column of A0 relative to ew . Let c00 ∈ RV (G) be defined by c00u = cu , u ∈ V (G) \ {v}, and c00w = cw − 1. Clearly (G00 , F ) ∈ C , thus A00 x00 ≥ c00 , x00 ≥ 0 has Chv´atal rank at most 1. It is not difficult to see that the point z¯00 defined by z¯h00 = z¯h0 , h ∈ E(G0 )\{ew }, and z¯e00w = 1−ze0 w is in the first closure of A00 x00 ≥ c00 , x00 ≥ 0. Furthermore, c00 (S) = c(S) is odd for every S ∈ L and z 00 (δG00 (S) \ F ) = 1, c00 (W ) = c(W ) − 1 is odd and z 00 (δG00 (W ) \ F ) = 1. Thus z 00 satisfies at equality |E(G00 )| linearly independent inequalities, therefore it is a fractional vertex of the first closure of A00 x00 ≥ c00 , x00 ≥ 0, a contradiction. ¦

4.4 Proof of Theorem 4.1

73

Claim 4.7. If G is bipartite, G \ F is connected, and L(G) ∩ F = ∅, then x¯ is half-integral. Proof of claim. Since all the cycles in the graph are even, by a theorem of Heller and Tompkins [24], the nodes in G can be partitioned into two subsets R, B such that, for every e = vw ∈ E(G), v and w are in the same class of − + − the partition if and only if σv,e 6= σw,e . Let L+ R and LR (resp. LB and LB ) be the sets of loops of G with respectively a +1 and a −1 sign, incident with nodes in R (resp. in B). By symmetry, we may assume c(R) ≥ c(B). If V (G) ∈ / L , then by Claim 4.6 x¯ is half-integral. Thus we assume that V (G) ∈ L . Then the odd-cut inequality relative to V (G), that is P − + − xl ≥ 1, is satisfied tightly by x ¯ and it is linearly independent l∈L+ R ∪LR ∪LB ∪LB from the equations in the system A(G, F )x = c. By summing together all the equalities in A(G, F )x = c corresponding to nodes in R and all the equalities in −A(G, F )x = −c corresponding to nodes in B we get X X xl − xl = c(R) − c(B). − l∈L+ R ∪LB

+ l∈L− R ∪LB

Since P c(V (G)) is odd, c(R) P − c(B) is odd, P hence c(R) − c(B) ≥ 1. Hence 1 = + − + − x ¯l ≥ l∈L+ ∪L− x¯l − l∈L− ∪L+ x¯l ≥ 1, because x¯ ≥ 0. Thus Pl∈LR ∪LR ∪LB ∪LB P R B P R B + + − + − x + − x + x ¯ = ¯ − ¯l , therefore L− l l R ∪ LB = ∅, l∈LR ∪LR ∪LB ∪LB l∈LR ∪LB l∈L− R ∪LB because x¯ > 0. But then the inequality defined by V (G) is linearly dependent from A(G, F )x = c, a contradiction. ¦ The following claim will be useful in the remaining of the proof. Claim 4.8. If x¯e = 12 for every e ∈ E(G), then (G, F ) satisfies the conditions a) and b) of Remark 4.14. x = c, Proof of claim. Since x¯e = 12 for every e ∈ E(G) and x¯ satisfies A(G, F )¯ x¯(δG (v) \ F ) is an integer for every v ∈ V , hence |δG (v) \ F | is even and a) is satisfied. ¯ of G \ F such that L(G) ¯ = ∅, then Given a connected component G ¯ ¯ ¯ δG (V (G)) \ F = ∅, hence c(V (G)) is even, otherwise V (G) defines an odd-cut inequality violated by x¯. Notice that, since A(G, F )¯ x = c, ¯ =1 c(V (G)) 2

X

(σv,vw +σw,vw )+

¯ vw∈E(G)

X

(σv,vw +σw,vw )+

vw∈F \L(G) ¯ v,w∈V (G)

X ¯ vw∈δG (V (G)) ¯ v∈V (G)

σv,vw

74

A class of matrices arising from bidirected graphs

¯ contribute 0 to the right-hand-side of the latter expresEven edges of E(G) ¯ contributes ±1, edges in F \ L(G) with both sion, each odd edge of E(G) ¯ ¯ contribute endnodes in V (G) contribute 0 or ±2, while edges in δG (V (G)) ¯ is congruent modulo 2 to ±1. Hence the number of odd edges in E(G) ¯ |δG (V (G))|. ¦

4.4.2

Structure of (G, F )

Given a cycle C and a family {ei , i ∈ I} of chords of C, we say that {ei , i ∈ I} is a family of non-crossing chords of C if for every pair of chords ei , ej , i, j ∈ I, each path in C between the two endnodes of ei , contains either both the endnodes of ej or none of them. Claim 4.9. F \ L(G) does not contain two disjoint edges with all endnodes in the same block of G \ F . Proof of claim. Otherwise let f = vw and f 0 = v 0 w0 be such edges. We show that (G, F ) contains a cycle C in G \ F such that f and f 0 are non-crossing cords of C. Since v, w, v 0 , w0 are distinct and in the same block of G \ F , such block is 2-connected. Let P1 be a path in G \ F from v to v 0 that does not pass through w. Notice that, by switching v 0 with w0 we can assume that P1 does not pass through w0 . Now let P2 be a path in G \ F from w0 to w that does not pass through v. Notice that P2 does not intersect P1 , as otherwise we get G4 as a minor. Now let P3 be a path in G \ F from w to v that does not pass through v 0 . Notice that P3 does not intersects P1 or P2 , as otherwise we get G4 as a minor. Now let P4 be a path in G \ F from v 0 to w0 that does not pass through v. Notice that P4 does not intersects P1 , P2 , or P3 as otherwise we get G4 as a minor. Hence C = v, P1 , v 0 , P4 , w0 , P2 , w, P3 , v is a cycle in G \ F , and f and f 0 are non-crossing cords of C. Now we show that the edges in F \ L(G) form a family of non-crossing chords of C. Let f 00 ∈ / {f, f 0 } be an edge in F \ L(G). Notice that f 00 is a chord of C, as otherwise by considering a shortest path from an endnode of f 00 to a node in C, we get G4 as a minor. Thus let C 00 be a subpath of C from one endnode of f 00 to the other. Clearly, if an internal node of C 00 is adjacent to another edge f 000 ∈ F \ L(G), then C 00 must contain also the other endnode of f 000 , as otherwise by deleting one endnode of f 000 we get G4 as a minor. Thus the edges in F \ L(G) form a family of non-crossing chords of C. This implies that G \ F is connected. Now we show that there are no loops in F . If not, let l ∈ F ∩ L(G). By considering a shortest path from the endnode of l to a node in V (C), we get the minor G4 .

4.4 Proof of Theorem 4.1

75

We also show that G is bipartite. If not, let C 0 be an odd cycle, let (G , F 0 ) be obtained from (G, F ) by contracting the edges in C 0 , and let l be the new loop in F 0 obtained from the contraction of C 0 . By considering a shortest path from the endnode of l to a node in V (C), we get the minor G4 . Moreover, for every edge f ∈ F , the endnodes of f form a cutset for G. Otherwise, by symmetry assume that {v, w} is not a cutset of G. Thus there exists a shortest path P from an internal node of P3 to an endnode of f 0 , that does not pass through v and w. By considering the cycle v, w, P3 , the path P , and the edge f 0 , we get G4 as a minor. Hence by Claim 4.7, x¯ is half-integral. Thus by Claim 4.8 (G, F ) satisfies conditions a) and b) of Remark 4.14. Let T be a family of trails obtained by partitioning the edges in E(G) \ (E(C) ∪ F ) into trails that start and end in loops in E(G) \ F and/or into closed trails. Notice that T 0 = T ∪ {C} satisfies conditions (C1)-(C3) of Lemma 4.16. Hence by Lemma 4.16 there exists a balanced bipartition R, B of E(G). Now let y = x¯ + 1/2χ(R) − 1/2χ(B) and let z = x¯ − 1/2χ(R) + 1/2χ(B). y and z are integral, they satisfy (4.2) and x¯ = 1/2(y + z), a contradiction. ¦ 0

Let f = vw, f 0 = v 0 w0 be two edges in F \ L(G) such that v, w, v 0 , w0 are in the same connected component of G \ F . We say that f 0 is nested in f if every path in G \ F from v to w contains the nodes v 0 , w0 . We say that f and f 0 are nested if f 0 is nested in f or f is nested in f 0 . Claim 4.10. Let f = vw, f 0 = v 0 w0 be two edges in F \ L(G) such that v, w, v 0 , w0 are in the same connected component of G \ F . Then one of the following holds: (i) f and f 0 are adjacent, say v = v 0 , and for any two distinct nodes s, t ∈ {v, w, w0 } there exists a path in G \ F between s and t that does not pass through {v, w, w0 } \ {s, t}; (ii) f and f 0 are nested; (iii) one among v and w, say v is a cutnode of G \ F separating w from {v 0 , w0 } \ {v}. Proof of claim. By contradiction assume that none of the cases above applies to f, f 0 . At first we show that f and f 0 are not adjacent. Otherwise assume v = v 0 . By Claim 4.4, w 6= w0 . Since we are not in case (i), by symmetry either there is not a path in G \ F from v to w that does not pass through w0 , or there is not a path in G \ F from w to w0 that does not pass through

76

A class of matrices arising from bidirected graphs

v. The first case cannot happen because otherwise f 0 is nested in f , thus case (ii) applies. The second case cannot happen because otherwise v is a cutnode of G \ F separating w from {v 0 , w0 } \ {v}, thus case (iii) applies. Thus all the nodes v, w, v 0 , w0 are pairwise disjoint. Since v, w are in the same connected component of G \ F , and since f and f 0 are not nested, there is a path P from v to w in G \ F that does not contain both v 0 and w0 . Clearly P does not contain any node among v 0 and w0 , as otherwise we get G4 as a minor by deleting the endnode of f 0 that does not appear in P . In the same way let P 0 be a path from v 0 to w0 in G \ F that does not contain any node among v and w. Now let S be a path in G \ F with an endnode in P an the other endnode in P 0 such that all its internal nodes are not in V (P ) ∪ V (P 0 ). Clearly one endnode of S is an endnode of f , say v, and the other endnode of S is an endnode of f 0 , say v 0 . In fact otherwise by symmetry one endnode of S is an internal node of P . By contracting all the edges in S, some edges in P 0 , and by deleting one endnode of f 0 we get G4 as a minor. Now we show that any other path S 0 in G \ F with an endnode in P , and the other endnode in P 0 such that all its internal nodes are not in V (P ) ∪ V (P 0 ), has endnodes v, v 0 . We show this by contradiction. In fact, if its endnodes are w, v 0 or v, w0 , then we get G4 by deleting respectively w0 or w, and by contracting some edges. Otherwise, if its endnodes are w, w0 , then V (S) ∩ V (S 0 ) = ∅, otherwise we get G4 as a minor. Hence f and f 0 are contained in the same block of G \ F , but this contradicts Claim 4.9. Thus case (iii) applies, a contradiction. ¦ Claim 4.11. Let f = vw be an edge in F \ L(G) with both endnodes in a ¯ of G \ F , and let f 0 be an edge in F with exactly an connected component G ¯ Then one of the following holds: endnode v 0 ∈ V (G). (i) f and f 0 are adjacent; (ii) one among v and w, say v is a cutnode of G \ F separating w from v 0 . Proof of claim. By contradiction assume that none of the cases above applies to f, f 0 . Since f and f 0 are not adjacent, v 0 ∈ / f . Since case (ii) does not apply, there exists a path Pw in G \ F from w to v 0 that does not contain v. Symmetrically, there exists a path Pv in G \ F from v to v 0 that does not contain w. By considering the edge f , and the paths Pv and Pw , we get G4 as a minor, a contradiction. ¦ Claim 4.12. Let f = vw be an edge in F \ L(G) with both endnodes in a ¯ of G \ F , and let C be an odd cycle in G. ¯ Then one connected component G among v and w, say v is a cutnode of G \ F separating w from V (C) \ {v}.

4.4 Proof of Theorem 4.1

77

Proof of claim. By contradiction assume that the Claim does not hold. Since v is not a cutnode of G \ F separating w from V (C) \ {v}, then there exists a path Pw in G \ F from w to a node in V (C) \ {v} that does not contain v. Symmetrically, there exists a path Pv in G\F from v to a node in V (C)\{w} that does not contain w. Notice that C does not contain v or w, as otherwise we get an odd cycle containing f by using the paths Pv , Pw and the edge f . Thus V (C) ∩ {v, w} = ∅. Let (G0 , F 0 ) be the obtained from (G, F ) by contracting all the edges of C. Let l be the new loop in F 0 obtained from the contraction of F , and let v 0 be the endnode of l. By considering the edge f , the loop l, and the paths from the endnodes of f to v 0 in (G0 , F 0 ) corresponding to Pv and Pw , we get G4 as a minor, a contradiction. ¦ Given a subset ∆ of F , let Gsplit(∆) be obtained from G by deleting all the edges in ∆ and by adding, for every node v ∈ V (G) that was incident with sign +1 if and only if P at least one removed edge, a loop lv in v, with split(∆) ¯f ≥ 0, with sign −1 otherwise. Let ∆ be the set of these f ∈∆ σv,f x split(∆) split(∆) split(∆) new loops in G and let F = F ∩ E(G ) ∪ ∆split(∆) . Let split(∆) split(∆) split(∆) split(∆) E(Gsplit(∆) ) A = A(G ,F ). Let x¯ ∈R be obtained from x¯ by removing the components corresponding P to the edges in ∆, and by split(∆) setting, for every loop lv in ∆split(∆) , x¯lv = f ∈∆ σv,f x¯f . Claim 4.13. Let ∆ ⊆ F be such that ∆ \ L(G) 6= ∅, the graph induced by ∆ is connected, and (Gsplit(∆) , F split(∆) ) does not contain G4 as a minor. Then: (i) ∆ ∩ L(G) = ∅; (ii) G \ ∆ is connected; (iii) If ∆ is a star centered at a node v, then x¯split(∆) = λy + (1 − λ)z, where 0 < λ < 1, where y, z are integral and satisfy Asplit(∆) x0 = c, x0 ≥ 0, x0e ≤ 1, ∀e ∈ E(Gsplit(∆) ) \ {lv }. Moreover for every set U ∈ L , |δ(U ) \ F | = 2. (iv) If ∆ = {f }, with f = vw ∈ F \ L(G), then x¯ is half-integral. Proof of claim. Notice that (Gsplit(∆) , F split(∆) ) is in C . Since |V (Gsplit(∆) )| = |V (G)|, and |E(Gsplit(∆) ) \ L(Gsplit(∆) )| < |E(G) \ L(G)|, it follows that the system Asplit(∆) x0 = c, x0 ≥ 0 has Chv´atal rank at most 1. Notice that Asplit(∆) is obtained from A by removing the columns relative to the edges in ∆, and by adding columns, relative to the loops in ∆split(∆) , where the

78

A class of matrices arising from bidirected graphs

column relative to a loop lv incident with a node v is zero everywhere except the entry relative to v with value 2σv,lv . Notice that the space spanned by the columns of Asplit(∆) contains the space spanned by the columns of A. We show that there are |E(G)| linearly independent inequalities valid for the first closure of Asplit(∆) x0 = c, x0 ≥ 0 tight at x¯split(∆) . Let αs x ≥ β s , s = 1, . . . , t, be the odd-cuts inequalities for (4.2) tight at x¯. Since x¯ is a vertex and 0 < x¯ < 1 (by Claim 4.3), the system Ax = c, αs x ≥ β s , s = 1, . . . , t, has rank |E(G)|. Notice that Asplit(∆) x¯split(∆) = c, αs x¯split(∆) = β s , s = 1, . . . , t. Furthermore, since the space spanned by the columns of Asplit(∆) contains the space spanned by the columns of A, the system Asplit(∆) x0 = c, αs x0 ≥ β s , s = 1, . . . , t has rank at least |E(G)|. Therefore |E(Gsplit(∆) )| ≥ |E(G)|, thus |∆split(∆) | ≥ |∆|. Also, since the graph induced by ∆ is connected, by Claim 4.4, ∆\L(G) forms a tree. Hence |∆split(∆) | = |∆ \ L(G)| + 1, so |L(G) ∩ ∆| ≤ 1. Since every tree has at least split(∆) two leafs, there exists a loop l in ∆split(∆) such that x¯l is fractional. (i): If there is one loop in ∆ ∩ L(G), then |E(G)| = |E(Gsplit(∆) )|. Hence x¯ is a fractional vertex of the first closure of Asplit(∆) x0 = c, x0 ≥ 0, a contradiction by our minimality assumption. Therefore there are no edges in ∆ ∩ L(G). split(∆)

(ii): If G\∆ is not connected, let G0 be a connected component of Gsplit(∆) and let G00 be the union of all the other connected components of Gsplit(∆) . Let F 0 = F split(∆) ∩ E(G0 ), F 00 = F split(∆) ∩ E(G00 ), ∆0 = ∆split(∆) ∩ E(G0 ), ∆00 = ∆split(∆) ∩ E(G00 ). Let A0 = A(G0 , F 0 ) and A00 = A(G00 , F 00 ). Let x¯0 (resp. x¯00 ) be the restriction of x¯split(∆) to the edges of G0 (resp. G00 ). Let c0 and c00 be the restriction of c to G0 and G00 respectively. Notice that by Claim 4.5, E(G0 ) \ ∆split(∆) 6= ∅, E(G00 ) \ ∆split(∆) 6= ∅, hence both x¯0 and x¯00 have at least a fractional component. Consider the systems A0 x0 = c0 , x0 ≥ 0 A00 x00 = c00 , x00 ≥ 0

(4.11) (4.12)

Clearly (G0 , F 0 ) and (G00 , F 00 ) are in C . Since |V (G0 )| < |V (G)| and |V (G00 )| < |V (G)|, both systems (4.11) and (4.12) have Chv´atal rank at most 1. Since x¯ is a vertex of the first closure of (4.2), there are |E(G)| linearly independent inequalities valid for the first closure tight at x¯. By Lemma 4.9, and since x¯ > 0, such inequalities are either in the system Ax = c, or they are odd-cut inequalities for (4.2). By Lemma 4.9, given an irredundant odd-cut inequality, say relative to some set U ⊆ V (G), U cannot be partitioned into two nonempty sets U1 and U2 such that every edge between U1 and U2 is in F .

4.4 Proof of Theorem 4.1

79

Therefore either U ⊆ V (G0 ) or U ⊆ V (G00 ), and the corresponding odd-cut inequality is valid either for the first closure of (4.11) or (4.12), respectively. Notice that each equation of Ax = c corresponding to a node incident with no edge in ∆ is an equation of either A0 x0 = c0 or A00 x00 = c00 . Moreover, to each equation of Ax = c corresponding to a node v incident with some edges in ∆, corresponds an equation of either A0 x0 = c0 or A00 x00 = c00 which is linearly independent from all the other equations of Ax = c and all the odd-cut inequalities of either (4.11) or (4.12), since it is the only one that has a coefficient different from zero for the loop in ∆0 or ∆00 incident with v. Therefore the |E(G)| inequalities valid for the first closure of either (4.11) or (4.12), that correspond to the |E(G)| linearly independent inequalities valid for the first closure of (4.2) tight at x¯, are all linearly independent. Since ∆ is connected, |E(G)| ≥ |E(G0 )| + |E(G00 )| − 1. Therefore either x¯0 is a vertex of the first closure of (4.11), or x¯00 is a vertex of the first closure of (4.12), then either x¯0 or x¯00 is integral. In particular, x¯e = x¯0e for an edge e ∈ E(G0 ) \ ∆0 or x¯e = x¯00e for an edge e ∈ E(G00 ) \ ∆00 is integer, contradicting Claim 4.3. (iii): Now assume that ∆ is a star centered at node v ∈ V (G). Notice that x¯split(∆) satisfies the system Asplit(∆) x0 = c, x0 ≥ 0, x0f ≤ 1, ∀f ∈ F split(∆) \ {lv },

(4.13)

that has Chv´atal rank at most 1 by Lemma 4.10. By Lemma 4.8, the odd-cut inequalities for the latter system are the odd-cut inequalities for Asplit(∆) x0 = c, x0 ≥ 0. Moreover |E(G)| = |E(Gsplit(∆) )|−1. Hence x¯split(∆) is contained in a face of dimension 1 of the first closure of Asplit(∆) x0 = c, x0 ≥ 0, x0f ≤ 1, ∀f ∈ F split(∆) \ {lv }, thus it is a convex combination of two integral vertices, y and z, satisfying the latter system, i.e. x¯split(∆) = λy + (1 − λ)z where 0 < λ < 1. Moreover, by Claim 4.3, each edge in E(Gsplit(∆) ) \ F split(∆) is in δ(U ) for an odd-cut inequality of the latter system defined by a set U , tight at x¯split(∆) . It follows that every edge in E(Gsplit(∆) ) \ F split(∆) = E(G) \ F is in δ(U 0 ) for an odd-cut inequality of (4.13) defined by a set U 0 , tight at x¯split(∆) . It follows that ye , ze ∈ {0, 1} for every e in E(Gsplit(∆) ) \ {lv } and so y and z satisfy the system Asplit(∆) x0 = c, x0 ≥ 0, x0e ≤ 1, ∀e ∈ E(Gsplit(∆) )\{lv }. Since also y and z satisfy (4.13), and satisfy tightly all the odd-cut inequalities of (4.13) satisfied tightly by x¯split(∆) , it follows that for every set U ∈ L , |δ(U ) \ F | = 2. (iv): Now assume ∆ = {f }, with f = vw ∈ F \ L(G), and let fv and fw be the loops in F split(f ) incident with v and w respectively. In this case split(f ) split(f ) notice that x¯fv = x¯fw = x¯f . By applying the argument in (iii) to split(f ) 0 0 the system A x = c, x ≥ 0, x0f ≤ 1, ∀f ∈ F split(f ) , it follows that

80

A class of matrices arising from bidirected graphs

x¯split(∆) = λy + (1 − λ)z, where 0 < λ < 1, where y, z are integral and satisfy Asplit(∆) x0 = c, x0 ≥ 0, x0 ≤ 1. By Claim 4.3, x¯f ∈ / Z. Hence we can assume yfv = 0, and zfv = 1. Notice that yfw = 1 and zfw = 0. If not, let y¯, z¯ ∈ ZE(G) be obtained from y, z, respectively, by replacing the two components relative to fv , fw with one component relative to f with value y¯f = yfv = yfw , z¯f = zfv = zfw , respectively. Then x¯ = λ¯ y + (1 − λ)¯ z , a contradiction. Hence split(f ) split(f ) split(f ) split(f ) x¯fv = 1 − λ and x¯fw = λ. Since x¯fv = x¯fw = x¯f , then λ = 1/2. split(f ) Hence x¯ is half-integral and so x¯ is half-integral. ¦ Claim 4.14. Let f = vw and f 0 = vw0 be distinct edges in F \ L(G) such that, for any two distinct nodes s, t ∈ {v, w, w0 } there exists a path in G \ F between s and t that does not pass through {v, w, w0 } \ {s, t}. Then v is a cutnode of G. ¯ is the subgraph of G induced by the node v and the conFurthermore, if G nected component of G \ v containing w and w0 , ¯ ∩ F is a star centered at v; (i) E(G) ¯ is bipartite; (ii) G ¯ \ F is connected. (iii) G Proof of claim. In this case, by applying Claim 4.10 to both f and f 0 , and by noticing that there are no cycles in F¯ by Claim 4.4, it follows that each ¯ different from f and f 0 , is either incident with v, or is edge in F \ L(G) in G nested in f and in f 0 . We show that there are no edges in F \ L(G) nested in f and in f 0 and not incident with v. Let f 00 = v 00 w00 be such an edge, f 00 is not adjacent to f or f 0 . Let v 00 be an endnode of f 00 such that there exists a path in E(G) \ F from v 00 to w that does not pass through w00 . Now let (G0 , F 0 ) be obtained from (G, F ) by removing all the edges incident with v and different from f and f 0 , and by contracting f . By considering a path in (G0 , F 0 ) from w to v 00 that does not pass through w0 and w00 and a path in (G0 , F 0 ) from w0 to v 00 that does not pass through w and w00 , we get G4 as a ¯ are incident with v. minor. Hence all the edges in F \ L(G) in G ¯ is incident By Claim 4.11, each edge in F with exactly an endnode in V (G) ¯ are even. Thus E(G)∩F ¯ with v. Moreover, by Claim 4.12, all the cycles in G ¯ is a star centered at v, and G is bipartite. ¯ ∩ F \ L(G) ¯ it By applying Claim 4.13 (ii) to the set of edges in E(G) ¯ \ F is connected. follows that G Now we show that v is a cutnode of G \ F . By contradiction assume that ¯ is the connected component of v is not a cutnode of G \ F . In this case G 0 G \ F that contains v, w, w . In particular ∆ is an edge-cutset of G, and

4.4 Proof of Theorem 4.1

81

(Gsplit(F ) , F split(F ) ) ∈ C . Let ∆ be the family of edges in F with exactly an ¯ Notice that the edges in ∆ are all incident with v. By endnode in V (G). ¯ = G. Since (Gsplit(F ) , F split(F ) ) does not contain G4 as a Claim 4.13 (ii), G minor, and the graph induced by F is connected, Claim 4.13 (i) implies that there are no loops in F . By Claim 4.7, x¯ is half-integral. Thus by Claim 4.8, (G, F ) satisfies conditions a) and b) of Remark 4.14. By Lemma 4.18, there exists a balanced bipartition R, B of the edges in E(G). Now let y = x¯ + 1/2χ(R) − 1/2χ(B) and let z = x¯ − 1/2χ(R) + 1/2χ(B). y and z are integral, they satisfy (4.2) and x¯ = 1/2(y + z), a contradiction. Thus v is a cutnode of G \ F . Since each edge in F with exactly an ¯ is incident with v, then v is also a cutnode of G. ¦ endnode in V (G) Claim 4.15. If f ∈ F \ L(G) has both endnodes in the same connected component of G \ F , at least one endnode of f is a cutnode of G \ F . ¯ be the connected Proof of claim. Let v and w be the endnodes of f , and let G component of G \ F containing v, w. By contradiction assume that v and w are not cutnodes of G \ F . If there is another edge f 0 ∈ F \ L(G) with both ¯ then by Claim 4.4, f 0 is not parallel to f . By Claim 4.10, endnodes in G, 0 f and f are in case (i),(ii), or (iii) of such Claim. Case (iii) cannot happen because v, w are not cutnodes of G \ F . Case (i) cannot happen, as otherwise we contradict Claim 4.14 (i) because v, w are not cutnodes of G \ F . Thus case (ii) applies, and since v and w are not cutnodes of G \ F , f 0 is nested in ¯ different from f . Hence all the edges in F \ L(G) with both endnodes in G f are nested in f . ¯ are By Claim 4.11, all the edges in F with exactly one endnode in G ¯ adjacent to f . Notice that such edges in F with exactly one endnode in G ¯ are of two types, the loops in G, and the edges in F \ L(G) with exactly one ¯ endnode in G. We now show that there are no edges in F with exactly one endnode in ¯ Otherwise let ∆ ⊆ F contain such edges and f . Notice that, since ∆\{f } G. is a cut of G in F , and by the structure of the edges in F with endnodes in ¯ then (Gsplit(∆) , F split(∆) ) does not contain G4 as a minor. By Claim 4.13 G, ¯ are not in L(G). Thus (i), all the edges in F with exactly one endnode in G we contradict Claim 4.13 (ii), since G \ ∆ is not connected. Thus G \ F is ¯ = G, and L(G) ∩ F = ∅. Let P be a path in G \ F , from v connected, G to w. By the above discussion, P passes through all the nodes of G incident with some edge in F . By applying Claim 4.13 (iv) to ∆ = {f }, x¯ is half-integral. Thus by Claim 4.8, (G, F ) satisfies conditions a) and b) of Remark 4.14. Moreover, by Claim 4.12, all the cycles in G \ F are even, thus G is bipartite.

82

A class of matrices arising from bidirected graphs

By Lemma 4.19, there exists a partition of the edges in E(G) \ F in one closed trail, if L(G) ⊆ F , or, if L(G) \ F 6= ∅, in |L(G) \ F |/2 trails such that their first and last edge are in L(G) \ F , such that one of the trails passes through all the nodes incident with edges in F . Notice that such partition satisfies conditions (C1)-(C3) of Lemma 4.16. Hence by Lemma 4.16 there exists a balanced bipartition R, B of E(G). Now let y = x¯ + 1/2χ(R) − 1/2χ(B) and let z = x¯ − 1/2χ(R) + 1/2χ(B). y and z are integral, they satisfy (4.2) and x¯ = 1/2(y + z), a contradiction. ¦ For any block B of G containing a cut in F and any connected component Q of B \ F , we define the graph HQ,B as follows. Its node set denoted by VQ,B is the set of nodes in Q incident with edges in δB (Q) \ L(Q), and two nodes v and w are adjacent in HQ,B if and only if there exists a path from v to w in Q all of whose intermediate nodes are not in VQ,B . Claim 4.16. Assume that G \ F is not connected, let B be a block of G containing a cut in F , and let Q be a connected component of B \ F . Let (G0 , F 0 ) be the pair obtained from (G, F ) by deleting all the nodes in V (G) \ V (Q). Then we have: (i) the nodes in VQ,B = {v1 , . . . , vk } can be ordered in such a way that vi is a cut-node of G0 that disconnects vi−1 and vi+1 for every i = 2, . . . , k−1; (ii) if vw is in δB (Q) \ L(Q) and v ∈ {v2 , . . . , vk−1 }, then {v, w} is a node-cutset of B separating v1 from vk . Proof of claim. First, we prove that HQ,B is a tree. Suppose by contradiction that HQ,B contains a cycle C. Let v, v 0 and v 00 be three nodes in C. By definition of HQ,B , there exists a path, say Pv,v0 , in Q from v to v 0 that does not contain v 00 . Similarly, there exists a path, say Pv0 ,v00 , in Q from v 0 to v 00 that does not contain v. In the same way, we define a path Pv,v00 . Let us denote by VC the set of nodes belonging to the paths Pv,v0 , Pv0 ,v00 , and Pv,v00 . Let w, w0 and w00 be nodes in B such that vw, v 0 w0 and v 00 w00 ∈ δB (Q) \ L(Q). From the definition of Q, it follows that these edges belong to F . Suppose that w, w0 and w00 are distinct nodes. Since B is not a cutedge of G, B is 2-connected. So we consider a shortest path P in B \ {v} from w to the node set (VC \ {v}) ∪ {w0 , w00 }. W.l.o.g, we may assume that the nodes v 0 and w0 are not in P (otherwise v 00 and w00 are not in P ). Let P 0 be the path vw, P, w00 v 00 if P ends in w0 , and vw, P otherwise. By deleting w0 and considering the subgraph consisting of a loop in F at v 0 and the paths Pv,v0 , Pv0 ,v00 and P 0 , we see that B contains the minor G4 , a contradiction. Similarly, if exactly two of the nodes w, w0 and w00 are identical, we can see the minor G4 .

4.4 Proof of Theorem 4.1

83

Now we may suppose that w = w0 = w00 . From Claim 4.13 (ii) it follows that there exists an edge (¯ v , w) ¯ ∈ δB (Q) \ L(Q), where v¯ ∈ Q and w¯ 6= w. Then consider a shortest path Pv¯ in Q from v¯ to the set VC . If Pv¯ ends either in v, in v 0 , or in v 00 , then we may assume that it ends in v 0 and by deleting w ¯ and considering the subgraph consisting of a loop in F at v¯ and the paths Pv¯, Pv,v0 , Pv0 ,v00 and the edges vw and v 00 w, we see that B contains the minor G4 , a contradiction. Otherwise, we may assume that Pv¯ ends in ¯ and considering a node of Pv,v0 different from v and v 0 . Then, by deleting w the subgraph consisting of a loop in F at v¯ and the paths Pv¯, Pv,v0 and the edges vw and v 0 w, we see the minor G4 , a contradiction. This terminates the proof that HQ,B is a tree. Now we show that the graph HQ,B is a path. Suppose by contradiction that HQ,B has a node v0 of degree at least 3. Let v, v 0 and v 00 be three neighbors of v0 in HQ,B . As above, we can define the paths Pv,v0 , Pv0 ,v00 and Pv,v00 , and obtain a contradiction. Hence, the graph HQ,B is a path. Let us prove statement (i). Let 2 ≤ i ≤ k − 1. Since HQ,B is a path, the sets of nodes {v1 , . . . , vi−1 } and {vi+1 , . . . , vk } are contained in two different connected components of Q \ {vi }, say K1 and K2 respectively. Suppose by contradiction that vi does not disconnect vi−1 and vi+1 in G0 . Then there exists a path in G0 \ {vi } joining K1 and K2 with at least one edge in F . As vi ∈ VQ,B , there exists an edge vi w ∈ δB (Q). By deleting w, we can see that B contains the minor G4 , a contradiction. Let us prove statement (ii). Let e = vw ∈ δB (Q) \ L(Q) for some v ∈ {v2 , . . . , vk−1 }. Suppose by contradiction that there exists a path P from v1 to vk in the graph B \ {v, w}. Let P 0 be a path in Q from v1 to vk . Statement (i) implies that there exists a cycle in P, P 0 containing v and an edge in F . By deleting w and considering the latter cycle, we see that B contains the minor G4 , a contradiction. ¦ Claim 4.17. Assume that G \ F is not connected, let B be a block such that B \ F is not connected, and let Q be a connected component of B \ F . Let VQ,B = {v1 , . . . , vk } be defined as in statement Claim 4.16 (i). Let V 00 be the set of nodes w ∈ V (Q) such that in Q there exist a path from w to v1 that does not pass through vk , and a path from w to vk that does not pass through v1 . Let (G00 , F 00 ) be the subgraph of G induced by the nodes in V 00 . Then: (i) for any 1 ≤ l, l0 ≤ k, there exists a path in B from vl to vl0 that does not contain any node in V (Q) \ {vl , vl0 }; (ii) each edge in F 00 \ L(G00 ) has endnodes in {v1 , vk } ∪ {v ∈ V 00 : v is a cutnode of G \ F separating v1 and vk };

84

A class of matrices arising from bidirected graphs

(iii) for each pair of edges f = vw, f 0 = v 0 w0 in F 00 \ L(G00 ), either f and f 0 are nested, or one among v and w, say v is a cutnode of G00 \ F 00 separating w from {v 0 , w0 } \ {v}; (iv) each edge in F \ WB with exactly one endnode in V 00 is incident with v1 or vk ; (v) all the cycles in G00 are even. Proof of claim. Let us prove (i). For every edge f = vi w ∈ δB (Q) \ L(Q), 2 ≤ i ≤ k − 1, we define Gf as the subgraph of G induced by the connected component of G \ {vi , w} containing v1 , and by the nodes {vi , w}. Since B is 2-connected, we observe that Gf is 2-connected. For any two distinct edges f = vi w and f 0 = vi0 w0 in δB (Q) \ L(Q), 2 ≤ i ≤ i0 ≤ k − 1, we have that Gf ⊂ Gf 0 or Gf 0 ⊂ Gf , where Gf ⊂ Gf 0 if i < i0 . Indeed, by Claim 4.16 (ii) applied to f and f 0 , either v1 and f are in Gf 0 in which case Gf ⊂ Gf 0 , or v1 and f 0 are in Gf in which case Gf 0 ⊂ Gf . Moreover, if Gf ⊂ Gf 0 , then {vi , w} is a node-cutset of Gf 0 separating v1 from any endnode of f 0 that is not an endnode of f . Let v1 w0 ∈ δB (Q) \ L(Q) for some node w0 . If k = 2, then since B is 2-connected there exists a path from w0 to vk that does not pass through v1 and so it contains no node in V (Q) \ v2 , which implies (i). Now assume k ≥ 3. Let f1 , . . . , fs be the edges in δB (Q) \ L(Q) incident with one of the nodes v2 , . . . , vk−1 . We may assume that Gfi ⊂ Gfi+1 for i = 1, . . . , s − 1. Let wi be the endnode of fi in G \ Q for i = 1, . . . , s. Notice that f1 = v2 w1 , fs = vk−1 ws and w0 ∈ Gf1 . Since Gf1 is 2-connected, there exists a path P0 from w0 to w1 in Gf1 \ v1 . Since v2 w1 is the unique edge in δB (Q) \ L(Q) and in Gf1 , P0 does not contain and any node in V (Q) ∪ {v2 }. Let vk ws+1 ∈ δB (Q)\L(Q). As before, we can show that there exists a path Ps from ws to ws+1 in G that does not contain any node in V (Q)∪V (Gfs )\{ws }. Now we show that for i = 1, . . . , s − 1, there exists a path Pi from wi to wi+1 in Gfi+1 not containing any node of Q. Let 1 ≤ i ≤ s − 1, vj , vj 0 such that fi = vj wi and fi+1 = vj 0 wi+1 . Since {vj , wi } is a node-cutset of Gfi+1 separating the node-set {v1 , w0 , . . . , wi−1 }\wi from {vj 0 , wi+1 }\{vj , wi }, notice that the subgraph H of G induced by the nodes in V (Gfi+1 ) \ V (Gfi ) ∪ {vj , wi } is 2-connected and fi and fi+1 are the only edges in E(H) ∩ (δB (Q) \ L(Q)). This implies that there exists a path Pi from wi to wi+1 in H not containing any node of Q. Therefore, for any 1 ≤ l, l0 ≤ k, choose wi and wi0 (0 ≤ i, i0 ≤ s + 1) such that vl wi , vl0 wi0 ∈ δB (Q)\L(Q). By considering the path vl , vl wi , Pi , . . . , Pi0 −1 , vl0 wi0 , vl0 , this proves (i).

4.4 Proof of Theorem 4.1

85

Now we show (ii). Let f = vw ∈ F 00 \ L(G00 ). Suppose first that v, w are both distinct from v1 . Then, by Claim 4.11 applied to the edges f, v1 w0 , we may assume that v is a cutnode of G00 \ F 00 separating v1 and w. If w = vk , then we are done. So assume w 6= vk . If w does not separate v and vk , then be deleting w and considering a path from v1 to vk that contains no node in V (Q) \ {v1 , vk } (by (i)), and a path from v1 to vk passing through v in Q \ w, we get the minor G4 , a contradiction. Thus w is a cut-node of G00 \ F 00 separating v and vk . It follows that v and w are cut-nodes of G00 \ F 00 separating v1 and vk . Now we may assume v = v1 and w 6= v1 , vk . By applying Claim 4.11 to the edges f, vk ws+1 , we have that w is a cut-node of G00 \ F 00 separating v1 and vk . Hence w is a cut-node of G \ F separating v1 and vk . This concludes the proof of (ii). Let us show (iii). We observe that case (i) of Claim 4.10 does not hold. In fact otherwise, we can assume by symmetry v = v 0 , then from (ii) it follows that one of the nodes v, w, w0 separates the other two in G00 \ F 00 , a contradiction. Then (iii) directly follows from Claim 4.10. We show that statement (iv) holds. Otherwise, let f be an edge in F \WB with exactly one endnode, say v, in V 00 , such that f is not incident with v1 or vk . Notice that by (i) there exists a path P1 in B from v1 to vk that does not contain any node in V (Q) \ {v1 , vk }. By considering P1 , and a path P2 in G \ F from v1 to vk that passes through v, we get the minor G4 . Finally, we prove (v). Suppose by contradiction that there exists an odd cycle C in G00 . If v1 , vk ∈ / V (C), then by contracting all the edges of C, we get a new loop in F , and now we can get the minor G4 as in the above proof of statement (iv). So we may assume by symmetry that v1 ∈ V (C). Let v ∈ V (C) \ {v1 }. By definition of G00 , there exists a path from v to vk in G00 \ F 00 that does not pass through v1 . We may assume that such a path does not contain any node in V (C) \ v (otherwise we consider some subpath). Then, using (i), we can find an odd cycle containing an edge in F , a contradiction. ¦

4.4.3

x¯ is half-integral

In the following two claims we show that x¯e = 1/2 for every e in E(G). In Claim 4.18 we show this in the case G \ F connected. In Claim 4.19 we show this in the remaining case G \ F not connected, and, for every maximal set V¯ of nodes connected in G \ F , we give the structure of the nodes in V¯ incident with edges in δ(V ) \ L(G). Claim 4.18. If G \ F is connected, x¯e = 1/2 for every e in E(G).

86

A class of matrices arising from bidirected graphs

Proof of claim. Assume that for every two edges f = vw and f 0 = v 0 w0 in F \ L(G), either f and f 0 are nested, or one among v and w, say v is a cutnode of G \ F separating w from {v 0 , w0 } \ {v}. Then then let f be an edge in F \ L(G) that is not nested in any other edge in F \ L(G). Notice that (Gsplit(f ) , F split(f ) ) does not contain G4 as a minor, hence, by Claim 4.13 (iv), x¯ is half-integral. Otherwise, by Claim 4.10, in G there are two edges f = v0 v1 and f 0 = v0 v2 in F \ L(G) such that for any two distinct nodes s, t ∈ {v0 , v1 , v2 } there exists a path in G \ F between s and t that does not pass through ¯ is the subgraph of G induced by the node v0 and {v0 , v1 , v2 } \ {s, t}. Let G ¯ \ v0 containing v1 and v2 . By Claim 4.14, v0 the connected component of G ¯ ¯ is bipartite. is a cutnode of G, E(G) ∩ F is a star centered at v0 , and G ¯ ∩ F . Notice that (Gsplit(∆) , F split(∆) ) is in C . Hence by Let ∆ = E(G) Claim 4.13 (iii), x¯split(∆) = λy 00 + (1 − λ)z 00 , where 0 < λ < 1, and where y 00 and z 00 are integral and satisfy

Asplit(∆) x00 = c, x00 ≥ 0, x00e ≤ 1, ∀e ∈ E(Gsplit(∆) ) \ {lv0 },

where lv0 is the edge in ∆split(∆) incident with v0 . Moreover for every set U ∈ L , |δ(U ) \ F | = 2. It follows that ye00 , ze00 ∈ {0, 1} for every e in E(Gsplit(∆) ) \ {lv0 } and so x¯00e ∈ {λ, 1 − λ} for every e in E(Gsplit(∆) ) \ {lv0 }. Hence x¯e ∈ {λ, 1 − λ} for every e in E(G), since for every edge in E(G) there exists an edge in E(Gsplit(∆) ) \ {lv0 } with its same value. If λ = 1/2 we are done, so assume E(G) λ , ye = 1 − ze = ½ 6= 1/2. Let y and z be the following vectors in {0, 1} 1 if x¯e = λ . By the above discussion, (Ay)s = (Az)s = cs for every 0 otherwise s 6= v0 . If V (G) ∈ / L , by Claim 4.6 x¯ is half-integral, hence we assume V (G) ∈ L and so |L(G) \ F | = 2, say L(G) \ F = {l1 , l2 }. We show that there exists an edge e¯ = vw in E(G) \ (F ∪ L(G)) such that there is only one set U in L with e¯ ∈ δ(U ). If not, by Claim 4.3, for every edge e ∈ E(G) \ (F ∪ L(G)) there are at least two sets U1e , U2e in L with e ∈ δ(U1e ), e ∈ δ(U2e ). Now consider the undirected graph Γ whose vertex set is E(G) \ F and where two elements e1 , e1 in V (Γ), that correspond to two edges e1 , e2 ∈ E(G) \ F , are adjacent in Γ if and only if there exists a set U ∈ L with e1 , e2 ∈ δ(U ) \ F . If Γ contains a cycle C, then the odd-cuts in L that correspond to the edges

4.4 Proof of Theorem 4.1 of Γ in C can be written in the form  1 1 0 0 ···  0 1 1 0 ···   .. . . . . .. .. ...  . .   0 ··· 0 1 1   0 ··· 0 0 1 1 0 ··· ··· 0

87

0 0 .. .



     0   1  1

  x1 ..  =  .   xk

 1 ..  , .  1

where k is the number of edges in C. Since all the odd-cut inequalities corresponding to sets in L are linearly independent, k is odd, and the only solution of this system is x1 = · · · = xk = 1/2, hence λ = 1/2 and x¯e = 1/2 for every e ∈ E(G). Hence we assume that Γ does not contain any cycle. Notice that Γ has no loops and no parallel edges. Note that the edgenode incidence matrix of Γ is the constraint matrix of the system of odd-cut inequalities corresponding to elements in L . Notice that by assumption all the nodes of Γ except the two corresponding to l1 , l2 have degree at least two, and the nodes corresponding to l1 , l2 have degree at least one by Claim 4.3, then Γ is a path from the node of Γ corresponding to l1 to the node of Γ corresponding to l2 . But, since V (G) ∈ L , the nodes of Γ corresponding to l1 and l2 are adjacent, hence Γ contains only one edge. Hence L = {V (G)}. Since G \ F is connected, and since there exists U ∈ L such that e ∈ δ(U ) for every e ∈ E(G) \ F (by Claim 4.3), then G contains only one node. In this case it is clear that A(G, F ) has the Edmonds-Johnson property, a contradiction. Now notice that, by switching signs on the endnodes of e¯, we can assume that σv,¯e 6= σw,¯e . Now let (G0 , F 0 ) be obtained from (G, F ) by contracting e¯, and let r be the node obtained from the contraction of e¯. Let A0 = A(G0 , F 0 ). Let x¯0 be the vector obtained from x¯ by removing the component corresponding to the removed edges, and let c0 be obtained from c by removing the components corresponding to v and w and adding a component relative to r with value c0r = cv + cw . Since (G0 , F 0 ) is in C , and |V (G0 )| < |V (G)|, then the system A0 x0 = c0 , x0 ≥ 0 has Chv´atal rank at most 1. Notice that x¯0 satisfies the system A0 x0 = c0 , x0 ≥ 0, and that the equation (A0 x0 )r = c0r is the sum of (Ax)v = cv and (Ax)w = cw . Furthermore, the odd-cut inequalities for A0 x0 = c0 , x0 ≥ 0 are exactly the odd-cut inequalities for (4.2) relative to sets U ⊆ V (G) that either contain both v and w or none of them. Hence x¯0 is in the first closure of A0 x0 = c0 , x0 ≥ 0, x0f ≤ 1 for every f ∈ F 0 , which has Chv´atal rank at most 1 by Lemma 4.10. Since all the odd-cut inequalities corresponding to sets in L are of this form except for one, and since x¯e¯ > 0, then x¯0 satisfies at equality |E(G)| − 2 = |E(G0 )| − 1 linearly independent

88

A class of matrices arising from bidirected graphs

inequalities among A0 x0 = c0 , x0 ≥ 0, x0f ≤ 1 for every f ∈ F 0 and the odd-cut inequalities corresponding to sets in L . Hence there exist two integral vertices of the first closure of the system 0 0 A x = c0 , x0 ≥ 0, x0f ≤ 1, f ∈ F 0 such that x¯0 = λ0 y 0 + (1 − λ0 )z 0 , where 0 < λ0 < 1. Notice that by Claim 4.3, ye0 , ze0 ∈ {0, 1} for every e in E(G). Hence by possibly switching y 0 with z 0 we can assume λ0 = λ. This implies that for every e ∈ E(G0 ), ye0 = ye , ze0 = ze . Hence (Ay)s = (Az)s = cs for every s ∈ / {v, w}, (A0 y 0 )r = (Ay)v + (Ay)w = (Az)v + (Az)w = cv + cw . Without loss of generality we can assume that v 6= v0 . But since we know that (Ay)s = (Az)s = cs for every s 6= v0 , hence (Ay)w = cv + cw − (Ay)v = cw . Hence both y and z are integral, satisfy the system Ax = c, x ≥ 0, and x¯ = λy + (1 − λ)z, a contradiction. ¦ Claim 4.19. If G \ F is not connected, then x¯e = 1/2 for every e in E(G). Proof of claim. We denote by Q1 , . . . , Ql the connected components of G \ F that have nodes in B. By statement Claim 4.16 (i), let us denote by Pi a path in Qi containing all nodes in VQi ,B and having its end-nodes, say zis and zit , in VQi ,B . We now show that there exists an edge in F whose end-nodes are equal 0 to zik and zik0 respectively, for some 1 ≤ i, i0 ≤ l, i 6= i0 and k, k 0 ∈ {s, t}. Suppose that this is not true. For any edge f = zik w ∈ δB (Qi ) for some 1 ≤ i ≤ l and k ∈ {s, t}, we define the subgraphs Kfs and Kft of B as follows. We know that w ∈ Qj for some j 6= i, w 6= zjs and w 6= zjt . Using Claim 4.16 (i), we denote by Kfs and Kft the connected components 0 of B\{zik , w} containing zjs and zjt respectively. Let f 0 = zik0 w0 such that min{|V (Kfs0 )|, |V (Kft 0 )|} = min {|V (Kfs )|, |V (Kft )|}. W.l.o.g, we may asf ∈δB (Qi ) 1≤i≤l (Kft 0 )| and

sume that |V (Kfs0 )| ≤ |V w ∈ Q1 . Since z1s ∈ Kfs0 is not adjacent 0 to zik0 and z1s ∈ VQ1 ,B , there exists an edge f 00 = z1s w00 ∈ Kfs0 for some w00 ∈ / Q1 . Then, we may assume that f 0 ∈ / Kfs00 (otherwise f 0 ∈ / Kft 00 ). Since 0 δB (Kfs0 ) ⊆ {zik0 , w0 }, it follows that V (Kfs00 ) $ V (Kfs0 ), contradicting the definition of f 0 . 0 Let f = zik zik0 for some 1 ≤ i, i0 ≤ l, i 6= i0 and k, k 0 ∈ {s, t}. We show that (Gsplit(f ) , F split(f ) ) does not contain G4 as a minor, which implies that x¯ is half-integral by Claim 4.13. 0 0 Let fik and fik0 be the new loops in Gsplit(f ) in zik and zik0 respectively. By contradiction assume that (Gsplit(f ) , F split(f ) ) contains G4 as a minor. Since (G, F ) does not contain G4 as a minor, by symmetry, we can assume that 0 the loop of G4 is fik . Then there exists a cycle C that passes through zik0 , a node v ∈ Qi incident with two edges in C not in F split(f ) , and a path P from

4.4 Proof of Theorem 4.1

89

zik to v that does not contain any node of C. Let VQi ,B = {v1 , . . . , vr }. We 0 may assume zik = v1 . Since zik0 ∈ / Qi and v ∈ Qi , there exist two nodes vl and 0 vl0 in VQi ,B ∩ C, 1 < l < l ≤ r, vl 6= v and vl0 6= v, such that the paths in C 0 from v to vl and from v to vl0 not containing zik0 are contained in the graph obtained from (G, F ) by deleting all the nodes in V (G)\V (Qi ). Then vl does not disconnect v1 and vl0 in such graph, contradicting statement Claim 4.16 (i). ¦

4.4.4

Shrinkable pairs of edges

Assume that G \ F is not connected, and let W be the set of edges in F with endnodes in different components of G \ F . For each block B of G, let WB = W ∩ E(B). Two adjacent edges wu, wv ∈ WB , are consecutive if there is no edge rw ∈ WB such that {r, w} is a cutset of B separating u and v. Let uw, vw be two edges in W incident with w. Notice that, by switching signs on u, v and w we can assume that σu,uw = σv,vw = +1 and that at least one among σw,uw and σw,vw is equal to +1. We say that (G0 , F 0 ) is obtained from (G, F ) by shrinking uw and vw if V (G0 ) = V (G), E(G0 ) = E(G)\{uw, vw}∪{uv}, F 0 = F \{uw, vw}∪{uv}, where the signing σ 0 on the 0 0 edges of G0 is defined by σu,uv = σu,uw , σz,e = σz,e for every e ∈ E(G0 ) \ {uv}, 0 0 z ∈ e, and σv,uv = +1 if σw,uw 6= σw,vw , σv,uv = −1 if σw,uw = σw,vw . Notice 0 0 that each cycle C in G that contains uv is even, since the corresponding cycles in G obtained from C 0 by removing the edge uv and by adding the two 0 0 edges wu, wv are even, and since σu,uw + σw,uw + σv,vw + σw,vw ≡4 σu,uv + σv,uv . 0 0 0 0 Thus (G , F ) satisfies the cycles condition, and A(G , F ) is totally halfmodular. By shrinking uw and vw, (G0 , F 0 ) may contain the minor G4 , thus we say that two edges uw, vw in W are shrinkable if the graph obtained from (G, F ) by shrinking uw and vw does not contain G4 as a minor. Notice that, by shrinking uw and vw, for each block B 0 of G0 , the nodes of B 0 are all contained in the same block of G. Claim 4.20. If there exists a node w in a block B of G such that w is incident with at least two edges in WB , then there are two shrinkable edges in WB incident with w. Proof of claim. Let w ∈ B be a node incident with at least two edges in WB . Let wu, wv be two consecutive edges in WB , and let (G0 , F 0 ) be the pair obtained by shrinking wu, wv. Let B 0 be the block(s) corresponding to B in G0 . If (G0 , F 0 ) does not contain G4 as a minor we are done. Hence we assume that (G0 , F 0 ) contains G4 as a minor.

90

A class of matrices arising from bidirected graphs

Fact 1: In B 0 there exists a cycle C such that, up to switching the roles of u and v, then v, w ∈ V (C), u ∈ / V (C), v is incident with two edges in 0 E(C) \ F , and w is incident with at least one edge in E(C) ∩ F 0 . Moreover, {v, w} is a cutset of B. Proof of fact 1: Since (G0 , F 0 ) contains G4 as a minor, in G0 there is a cycle C that contains at least one edge in F 0 , a node c ∈ V (C) incident with two edges in E(C) \ F 0 , and a path P from c to a node d such that V (P ) ∩ V (C) = {c}, such that E(P ) ∩ F 0 = ∅, and d is incident with an edge f = dt (possibly t = d) in F 0 incident with no other node in V (C) ∪ V (P ). Since (G, F ) does not contain G4 as a minor, then uv ∈ E(C) ∪ {f }. We show that uv = f . Otherwise, suppose that uv ∈ E(C). In this case the nodes in V (C) are all in the block B of G. If w ∈ V (C) \ {c}, then the edges in C \ {uv} ∪ {uw, vw} form two cycles in G. Let C 0 be the one passing through c. Notice that E(C 0 ) ∩ F 6= ∅, c is incident with two edges in E(C 0 ) \ F , and V (C 0 ) ∩ (V (P ) ∪ {t}) = {c}. Thus (G, F ) contains G4 as a minor, a contradiction. Thus w ∈ V (P ) ∪ {t}. Let C¯ be the shortest subpath of C containing c as an internal node and with endnodes that are incident in G with edges in WB . Notice that such path exists because the nodes u, v are incident in G with edges in WB . Moreover, all the nodes in C¯ are in the same connected component of G \ F , since the two edges in C incident with c are not in F . Let u¯ (resp. v¯) be the endnode of C¯ in the subpath of C from ¯ be the graph c to u (resp. v) that does not pass through v (resp. u). Let G obtained from B by deleting all the nodes not in the connected component of G \ F containing c. Assume t ∈ / V (B). By Claim 4.17 (i), there exists a path S in B from u¯ ¯ \ {¯ to v¯ that contains no node in V (G) u, v¯}. Since t ∈ / V (B), then t ∈ / V (S). ¯ Hence by using S, C, and P , we see that (G, F ) contains G4 as a minor, a contradiction. Hence t ∈ V (B). Thus all the nodes in V (P ) ∪ {t} are in B, and all the ¯ as E(P ) ⊆ E(G) \ F . nodes in V (P ) are in G, ¯ then by Claim 4.16 (i) one among u¯, v¯, d Assume f ∈ WB . Since d is in G, ¯ is a cutnode of G separating the other two. But the only possibility is that ¯ separating u¯ and v¯. So P has length zero. d = c and d is a cutnode of G Since w ∈ V (P ) ∪ {t}, then w ∈ {d, t}. By Claim 4.16 (ii), {d, t} is a cutset of B separating u¯ and v¯, thus {d, t} separates u and v, but this contradicts the choice of wu, wv to be consecutive. ¯ Notice that, by Claim 4.16 (i), one Thus f ∈ / WB , hence dt ∈ E(G). ¯ separating the remaining two. Since among u¯, v¯ and w is a cutnode of G ¯ between v¯ (resp. u¯) and every node in V (P ) ∪ {t} that there is a path in G

4.4 Proof of Theorem 4.1

91

does not pass through u¯ (resp. v¯), then w is a cutnode separating u¯ and v¯. ¯ by Claim 4.17 (i) there exists a path T Therefore w = c. Since u¯, v¯ ∈ V (G), ¯ \ {¯ in B between u¯ and v¯ that does not contain any node in V (G) u, v¯}. Let 0 0 0 ¯ C be the cycle u¯, C, v¯, T, u in G. Then t ∈ / V (C ), w ∈ V (C ), w is incident with two edges in E(C 0 ) \ F , E(C 0 ) contains at least an edge in WB (because ¯ V (P ) ∩ V (C 0 ) = {w}, therefore C the inner nodes of T are in V (G) \ V (G)), and P form a G4 minor. Hence uv = f , and we can assume that v ∈ V (P ). Clearly w ∈ V (C), as otherwise by considering C, P , and the edge vw we see that (G, F ) contains G4 as a minor. Notice that all the nodes in V (C) ∪ V (P ) ∪ {u} are in B. If not, then C is not in B, thus by considering C, P , and the edge among vw we see that (G, F ) contains G4 as a minor. Moreover w is incident with at least one edge in E(C) ∩ F , as otherwise by considering C and the edge uw, (G, F ) contains G4 as a minor. Let C¯ be the shortest subpath of C containing c as an internal node and with endnodes that are incident in G with edges in WB . Notice that such path exists since E(P ) ⊆ E(G) \ F , and since v and w are in different ¯ All the components of B \ F , as vw ∈ WB . Let c0 , c00 be the endnodes of C. nodes in C¯ are in the same connected component of G\F , since the two edges ¯ be the graph obtained from G by in C incident with c are not in F . Let G ¯ deleting all the nodes not in the connected component of G \ F containing C. 0 00 ¯ By Claim 4.16 (i), one among c , c , v is a cutnode of G separating the other ¯ separating c0 two. The only possibility is that v = c, and v is a cutnode of G 00 and c . By Claim 4.16 (ii), this implies that {v, w} is a cutset of B separating P 0 and P 00 , where P 0 and P 00 are the two disjoint paths in C from v to w. This concludes the proof of Fact 1. Let zw ∈ WB such that {z, w} is a cutset of B. Let V1 be a maximal set of nodes connected in B \ {z, w}, and let B1 be the subgraph of B induced by V1 ∪ {z, w}. Let B2 be the subgraph of B induced by V (B) \ V1 . Fact 2: If there exist edges wr0 , wr00 ∈ / F in B1 , and B2 respectively, then given any two consecutive edges uw, vw ∈ WB , uw, vw are shrinkable. Proof of fact 2: Clearly r0 , r00 6= z, as zw ∈ WB . We first show that in this case, for every edge ws in WB , {w, s} is a cutset of B. We know that there exists at least another edge in WB incident with w different from zw. Since B is 2-connected, there exists a path P 0 (resp. P 00 ), from r0 (resp. r00 ) to z, that does not pass through w. Clearly P 0 is contained in B1 , and P 00 is contained in B2 .

92

A class of matrices arising from bidirected graphs

Let C¯ be the shortest subpath of z, P 0 , w, P 00 , z containing w as an internal node and with endnodes that are incident in G with edges in WB . Notice that such path exists since z and w are in different components of B \ F , as ¯ All the nodes in C¯ are in the zw ∈ WB . Let c0 , c00 be the endnodes of C. ¯ be the graph same connected component of G \ F , since wr0 , wr00 ∈ / F . Let G obtained from G by deleting all the nodes not in the connected component ¯ By Claim 4.16 (i), w is a cutnode of G ¯ separating the of G \ F containing C. other two. By Claim 4.16 (ii), for every edge ws in WB , {w, s} is a cutset of B. Therefore s is in V (P 0 ) ∪ V (P 00 ) \ {w}. Thus we can order the edges f1 = ww1 , . . . , fu = wwl in WB incident with w in such a way that for every i = 1, . . . , l − 1, the edges fi and fi+1 are consecutive. We define the sets U1 , . . . , Ul+1 ⊆ V (B) such that Ui contains the nodes v ∈ V (B) such that there is a path from v to wi that does not contain nodes in {w, w1 , . . . , wl } \ {wi }, and a path from v to wi−1 that does not contain nodes in {w, w1 , . . . , wl } \ {wi−1 } for every i = 1, . . . , l + 1, where w0 = wu+1 = w. Notice that we can assume r0 ∈ U1 and r00 ∈ Ul+1 , therefore c0 ∈ U1 and c00 ∈ Ul+1 . Next we show that, for every edge wy ∈ E(B) \ WB , y ∈ U1 ∪ Ul+1 . If ¯ Let s be the first not, suppose that y ∈ Ui , i ∈ {2, . . . , l}. Thus y ∈ G. node incident with edges in WB in a path from y to wi in the subgraph of ¯ hence, by Claim 4.16 (i), one G induced by Ui ∪ {wi , wi−1 }. Then s is in G, 0 00 ¯ among s, c , c is a cutnode of G separating the other two, a contradiction. Now let (G0 , F 0 ) be obtained from (G, F ) by shrinking two consecutive edges wwi , wwi+1 , 1 ≤ i ≤ l − 1. We show that (G0 , F 0 ) does not contain the minor G4 . Otherwise, by Fact 1 and by symmetry, there exists a cycle S in B 0 passing through wi and w and not through wi+1 such that wi is incident with two edges in E(S) \ F . Therefore all nodes in S are contained in Ui ∪ {wi−1 , wi }. Thus, i = 1. Let S 0 and S 00 be the two distinct subpaths of S between w and w1 . Then one among S 0 and S 00 , say S 0 , does not pass ¯ hence there exists through c0 . Since ww1 ∈ WB , then w1 is not in V (G) 0 0 an edge of S in WB . Let c be the node in a S incident with an edge in WB ∩ E(S 0 ) closest to w. By the choice of f1 , c 6= w. Notice that, between ¯ that does not pass any two distinct nodes among c, c0 , c00 there is a path in G through the third one. But by Claim 4.16 (i), one among c, c0 , c00 should be ¯ separating the remaining two nodes, a contradiction. This a cutnode of G concludes the proof of Fact 2. By Fact 2, we assume that in B1 there is no edge wr ∈ / F , r 6= z. Notice that we can choose the node z in such a way that in B1 there is no other edge wt ∈ WB such that {w, t} is a node cutset of B.

4.4 Proof of Theorem 4.1

93

Clearly, for every edge vw ∈ F with v ∈ B1 , then vw ∈ WB , otherwise we have an edge wr ∈ / F with r 6= z, and r ∈ V (B1 ). Assume that there exist u, v ∈ B1 such that uw, vw ∈ WB , u, v 6= z. Then we may choose uw, vw consecutive. If {u, w} or {v, w} is a cutset, then we contradict the minimality of B1 . So {u, w} and {v, w} are not cutsets, and the result follows from Fact 1. Thus there is at most one edge vw ∈ WB with v ∈ B1 . Then there is one, otherwise w has no other neighbor in B1 except for z, thus z is a cutnode of B, contradicting the fact that B is a block. Notice that, by Claim 4.13 (ii), there exists an edge e in E \ F incident with w, therefore e ∈ B2 . Now let (G0 , F 0 ) obtained from (G, F ) by shrinking zw, vw and assume that (G0 , F 0 ) contains G4 as a minor. Since in G \ vw every path from v to w passes through z, then by Fact 1, there exists a cycle C passing through z and w and not through v such that the two edges in C incident with z are in E \ F . Clearly C must be contained in B2 . Moreover, z and v are in the same connected component of G \ F , as otherwise (G, F ) contains G4 as a minor by considering the cycle C, and a path in B1 from z to v in B1 that contains edges in F . Since zw ∈ WB , each of the two disjoint paths in C from z to w contains an edge in WB . Let C¯ be the shortest subpath of C containing z as an internal node and with endnodes that are incident in G with edges in WB . Notice that such path exists since z and w are in different components of B \ F , as zw ∈ WB . ¯ All the nodes in C¯ are in the same connected Let c0 , c00 be the endnodes of C. 0 ¯ be the graph obtained from G component of G\F , since wr , wr00 ∈ / F . Let G by deleting all the nodes not in the connected component of G \ F containing ¯ Notice that there are three disjoint paths in G ¯ from z to, respectively, C. v, c0 , and c00 , where z 6= c0 , c00 , v. But this contradicts Claim 4.16 (i). ¦ Claim 4.21. In each block B of G, each node is incident with at most one edge in WB . Proof of claim. Otherwise there exists a node w in a block B of G such that w is incident with at least two edges in WB . By Claim 4.20, there are two shrinkable edges wu, wv in WB incident with w. Let (G0 , F 0 ) be the pair obtained by shrinking wu, wv, and let A0 = A(G0 , F 0 ). Notice that, by switching signs on u, v and w we can assume that σu,uw = σv,vw = +1 and that at least one among σw,uw and σw,vw is equal to +1. Now let c0 be defined by c0z = cz for every z ∈ V (G0 ) \ {v, w}, c0v = cv and c0w = cw if σw,wu 6= σw,wv , c0v = cv − 2 and c0w = cw − 2 otherwise. Now let x¯0e = 1/2 for every e ∈ E(G0 ). Since |V (G0 )| = |V (G)| and |E(G0 ) \ L(G0 )| < |E(G) \ L(G)|, then the system A0 x0 = c0 , x0 ≥ 0 has Chv´atal rank at most 1. We show that x¯0 is

94

A class of matrices arising from bidirected graphs

in the first closure of the system A0 x0 = c0 , x0 ≥ 0. Clearly x¯0 satisfies such system. Moreover, the odd-cut inequalities for A0 x0 = c0 , x0 ≥ 0 are satisfied by x¯0 , since the same inequalities are satisfied by x¯, since for every z ∈ V (G0 ), c0z has the same parity of cz , and since the edges in E(G) \ F are exactly the same edges in E(G0 ) \ F 0 . We show that x¯0 satisfies tightly |E(G0 )| = |E(G)|−1 linearly independent inequalities valid for the first closure of the system A0 x0 = c0 , x0 ≥ 0. Since x¯ is a vertex of the first closure of the system (4.2), it satisfies tightly |E(G)| linearly independent inequalities, αi x ≥ β i , i = 1, . . . , |E(G)|, valid for the first closure of (4.2). Let M be the |E(G)| × |E(G)| matrix, such that its i-th row is equal to αi . Now let M 0 be the |E(G)| × |E(G)| − 1 matrix obtained from M by substituting the two columns corresponding to the edges wu, wv ∈ E(G) with a new column corresponding to the new edge uv ∈ E(G0 ), which is the sum of the two removed columns if σw,wu 6= σw,wv and is the difference between the column corresponding to the edge uw and the one corresponding to edge vw otherwise. Notice that each row of M 0 corresponds to an inequality of the first closure of A0 x0 = c0 , x0 ≥ 0 satisfied tightly by x¯0 , and that M 0 has rank |E(G0 )| = |E(G)| − 1. Hence x¯0 is a fractional vertex of the first closure of A0 x0 = c0 , x0 ≥ 0, a contradiction. ¦

4.4.5

The end: finding a balanced bipartition

In the remainder of the chapter we show that there exists a balanced bipartition R, B for (G, F ). Notice that this will conclude the proof of Theorem 4.1. Indeed, the vectors y = x¯ + 21 χ(R) − 12 χ(B) and z = x¯ + 21 χ(B) − 21 χ(R) are integral, since x¯e = 12 for every e ∈ E(G), y and z satisfy the system (4.2), since R, B is a balanced bipartition, and x¯ = 21 (y + z), contradicting the fact that x¯ is a vertex of the first closure of (4.2). Notice that, since x¯e = 12 for every e ∈ E(G), by Claim 4.8, (G, F ) satisfies the conditions a) and b) of Remark 4.14. ˜ F˜ ) in C satisfying condiWe construct recursively a family G of pairs (G, tions a) and b) of Remark 4.14 as follows. Initially G = {(G, F )}. Until there ˜ F˜ ) ∈ G such that G ˜ has two distinct blocks B1 , B2 such that, for exists (G, i = 1, 2, either Bi contains an odd cycle or E(Bi ) ∩ F˜ 6= ∅, let w be a cutnode ˜ separating B1 and B2 , let (G ˜ 1 , F˜1 ) and (G ˜ 2 , F˜2 ) be obtained from (G, ˜ F˜ ) of G ˜ F˜ )} ∪ {(G ˜ 1 , F˜1 ), (G ˜ 2 , F˜2 )}. by breaking B1 and B2 at w, and let G := G \ {(G, ˜ F˜ ) of The process ends with a family G such that, for each element (G, ˜ F˜ ) has at most one block B such that B contains an odd cycle or G , (G, E(B) ∩ F˜ 6= ∅.

4.4 Proof of Theorem 4.1

95

˜ F˜ ) in G has a balanced bipartition, then By Lemma 4.15, if every (G, (G, F ) has a balanced bipartition. ˜ F˜ ) be an element of G . We show that (G, ˜ F˜ ) has a balanced Let (G, bipartition. ˜ \ F˜ is connected, then there exists a balanced bipartition Claim 4.22. If G ˜ of E(G). ˜ is either an edge of G, or Proof of claim. Notice that each edge in F˜ ∩ L(G) was introduced in the construction of G . ˜ Assume that for every two edges f = vw and f 0 = v 0 w0 in F˜ \ L(G), 0 either f and f are nested, or one among v and w, say v, is a cutnode ˜ \ F˜ separating w from {v 0 , w0 } \ {v}. Choose an edge f = vw in of G ˜ not nested in other edges in F˜ \ L(G). ˜ Assume that there is an F˜ \ L(G) 0 0 0 0 ˜ f 6= f , not nested in f . Then one among v and edge f = v w ∈ F˜ \ L(G), ˜ \ F˜ separating w from {v 0 , w0 } \ {v}. Since f is w, say v, is a cutnode of G ˜ w is a cutnode of G, ˜ contradicting the not nested in other edges in F˜ \ L(G), ˜ ˜ ˜ construction of G . Hence in G, all the edges in F \ L(G) are nested in f . ˜ are incident with v or w. Assume We show that all the edges in F˜ ∩ L(G) ˜ not adjacent to f . Then by that there is an edge f 0 = v 0 v 0 ∈ F˜ ∩ L(G) ˜ \ F˜ Claim 4.11 and 4.12, one among v and w, say v, is a cutnode of G 0 ˜ ˜ separating w from v . Since f is not nested in other edges in F \ L(G), v is ˜ contradicting the construction of G . Thus all the edges in a cutnode of G, ˜ ˜ F ∩ L(G) are incident with v or w. ˜ is bipartite. Assume that there is an odd cycle Similarly, we show that G ˜ Then by Claim 4.12, one among v and w, say v, is a cutnode of C in G. ˜ ˜ G \ F separating w from V (C) \ {v}. Since f is not nested in other edges ˜ then v is a cutnode of G, ˜ contradicting the construction of G . in F˜ \ L(G), ˜ is bipartite. Thus G ˜ \ F˜ from v to w. Clearly P passes through all the Let P be a path in G ˜ nodes of G incident with some edge in F˜ . ˜ \ F˜ in one By Lemma 4.19, there exists a partition of the edges in E(G) ˜ ⊆ F˜ , or, if L(G) ˜ \ F˜ 6= ∅, in |L(G) ˜ \ F˜ |/2 trails such that closed trail, if L(G) ˜ ˜ their first and last edge are in L(G) \ F , such that one of the trails, say T , passes through all the nodes incident with edges in F˜ . Let T be the family of trails obtained from the above partition of the ˜ \ F˜ , by adding in T all the edges in F˜ ∩ L(G). ˜ Clearly T edges in E(G) satisfies (C1)-(C3) of Lemma 4.16, thus there exists a balanced bipartition ˜ of E(G).

96

A class of matrices arising from bidirected graphs

˜ there are two edges f = vw and f 0 = vw0 in Thus, by Claim 4.10, in G ˜ such that for any two distinct nodes s, t ∈ {v, w, w0 } there exists a F˜ \ L(G) path in G \ F between s and t that does not pass through {v, w, w0 } \ {s, t}. We show that the edges in F˜ , form a star centered at v. Notice that, by ˜ has at most one block B such that B contains an the construction of G , G ˜ odd cycle or E(B) ∩ F 6= ∅. Thus by Claim 4.14 and by the construction of G , the edges in F˜ that were edges of G, form a star centered at v. Any ˜ Thus other edge l ∈ F˜ is obtained from the construction of G , and l ∈ L(G). 0 0 ˜ ˜ assume that l = v v is an edge in F ∩ L(G) not adjacent to f introduced in the construction of G . Then by applying Claim 4.11 and 4.12 to both f and ˜ \ F˜ separating w from v 0 . Since all the f 0 , it follows that v is a cutnode of G ˜ are incident with v, then v is a cutnode of G, ˜ contradicting edges in F˜ \ L(G) ˜ are incident with v or the construction of G . Thus all the edges in F˜ ∩ L(G) w. ˜ is bipartite. Assume that there is an odd cycle Similarly, we show that G ˜ Then by applying Claim 4.12 to f and f 0 , it follows that v is a C in G. ˜ \ F˜ separating w from V (C) \ {w}. Since all the edges in cutnode of G ˜ are incident with v, then v is a cutnode of G, ˜ contradicting the F˜ \ L(G) ˜ is bipartite. construction of G . Thus G ˜ Thus, by Lemma 4.18, there exists a balanced bipartition of E(G). ¦ ˜ \ F˜ is not connected. Let V¯ be a maximal set of nodes connected Thus G ˜ ˜ ¯ F¯ ) be obtained from (G, ˜ F˜ ) by deleting all the nodes in in G \ F . Let (G, ˜ \ V¯ . Notice that (G, ¯ F¯ ) is in C , and that it satisfies conditions a) and V (G) ¯ ∩ F¯ are of three types: b) of Remark 4.14. Notice that the loops in E(G) ¯ artificial loops created in the loops of F incident with some node in V (G); construction of the family G ; and loops corresponding to edges of F with one ¯ and one in V (G) ˜ \ V (G) ¯ obtained by deleting the endnode endnode in V (G) ˜ \ V (G). ¯ Let ∆ ⊆ F¯ be the set of loops in E(G) ¯ ∩ F¯ of the last type, in V (G) ¯ ¯ and let Θ be the set of the loops in E(G) ∩ F of the other two types. Notice ˜ By Claim 4.21 that we identify the loops in ∆ with the original edges in G. ¯ is incident with at most one edge in ∆, since all the edges in each node of G ˜ ∆ correspond to edges in the same block of G. ¯ incident with edges in Let v1 , . . . , vk be an ordering of the nodes in V (G) ∆ as in Claim 4.16 (i), and let P = v1 , P1 , v2 , . . . , vk−1 , Pk−1 , vk be a path in ¯ F¯ from v1 to vk , where Pi is a path from vi to vi+1 for every i = i, . . . , k−1. G\ Let li be the only loop in ∆ incident with vi for every i = 1, . . . , k. Notice that, by the construction of G , and by Claim 4.17, it follows that: ¯ is bipartite; • G

4.4 Proof of Theorem 4.1

97

¯ has endnodes in {v1 , vk }∪{v ∈ G ¯ : v is a cutnode • each edge in F¯ \L(G) ¯ \ F¯ separating v1 and vk }; of G • each edge in Θ is incident with v1 or vk . ¯ incident with some edge in F¯ . Thus P passes through all the nodes of G ¯ such Claim 4.23. There exists a balanced bipartition for the edges in E(G) that twoPloops li and li+1 P in ∆, are in the same side of the partition if and only if e∈{li ,E(Pi ),li+1 } v∈e σv,e ≡4 0, for every i = 1, . . . , k − 1. ¯ \ F¯ in Proof. By Lemma 4.19, there exists a partition of the edges in E(G) ¯ ⊆ F¯ , or, if L(G) ¯ \ F¯ 6= ∅, in |L(G) ¯ \ F¯ |/2 trails one closed trail, if L(G) ¯ ¯ such that their first and last edge are in L(G) \ F , such that one of the trails passes through all the nodes incident with edges in F¯ . Let T be such family of trails, and let T ∈ T be the one that passes through all the nodes incident with edges in F¯ . Let S be a shortest subtrail of T containing ¯ and since all the nodes v1 , . . . , vk . Since v1 , . . . , vk are all cutnodes of G ¯ ¯ P = v1 , P1 , v2 , . . . , vk−1 , Pk−1 , vk is a path in G \ F , it follows that S can be written in the form S = v1 , S1 , v2 , . . . , vk−1 , Sk−1 , vk . ¯ in Let S 0 be obtained from S by adding to S all the edges in F¯ ∩ L(G) 0 such a way that there exists a subtrail of S containing all the edges in ∆ and no edge in Θ. Notice that this can be easily done since the endnodes of S are v1 and vk , since all the edges in ∆ are incident with the nodes v1 , . . . , vk , and since all the edges in Θ are incident with v1 , vk . Let T 0 be obtained from T by replacing its subtrail S with S 0 . Let T 0 = T \ {T } ∪ {T 0 }. Clearly T 0 satisfies (C1)-(C3) of Lemma 4.16, thus there exists a balanced ¯ Moreover, by Remark 4.17, li and li+1 in ∆, are in the bipartition of E(G). P P same side of the partition if and only if e∈{li ,E(Si ),li+1 } v∈e σv,e ≡4 0, for every i = 1, . . . , k − 1. For every i = 1, . . . , k − 1, let Ei1 be the edges in the symmetric difference between E(Pi ) and E(Si ), and let Ei2 = E(Pi ) ∪ E(Si ) \ ˜ is in Ei1 form a union of cycles, thatPare balanced since G Ei1 . The edges P bipartite. Thus vw∈E 1 (σv,vw + σw,vw ) ≡4 0, and 2 vw∈E 2 (σv,vw + σw,vw ) ≡4 i i P P 0. Thus vw∈E(Si ) (σv,vw + σw,vw ) + vw∈E(Pi ) (σv,vw + σw,vw ) ≡4 0. Since P P (σv,vw + σw,vw ) is even, it follows that vw∈E(Si ) (σv,vw + σw,vw ) ≡4 vw∈E(P ) i P vw∈E(Pi ) (σv,vw + σw,vw ) for every i = 1, . . . , k − 1. li+1 in ∆, are in the same side of the partition if and only if P Thus li and P e∈{li ,E(Pi ),li+1 } v∈e σv,e ≡4 0, for every i = 1, . . . , k − 1. ˜ Claim 4.24. There exists a balanced bipartition of the edges in E(G).

98

A class of matrices arising from bidirected graphs

Proof of claim. Let W be the set of edges in F˜ with endnodes in different ˜ \ F˜ . Consider the pair (G ˜ split(W ) , F˜ split(W ) ). Notice that G ¯ components of G split(W ) split(W ) ˜ ¯ ¯ ˜ is a connected component of G , and that F = E(G) ∩ F . Notice that, since by Claim 4.21 there are not two adjacent edges in W , then by Claim 4.23 there exists a balanced bipartition, as in Claim 4.23, for ˜ split(W ) . Thus we also have a the edges in each connected component of G split(W ) ˜ balanced bipartition for the edges in G . We now show that we can combine all these bipartitions to get a balanced bipartition of the edges in ˜ E(G). ˜ 0 be obtained from G ˜ by contracting all the edges in E(G) ˜ \ F˜ and Let G by removing all the loops. Notice that there is a one-to-one relation between ˜ 0 and the edges in W in G. ˜ Now let T ⊆ E(G ˜ 0 ) be a spanning the edges of G ˜ 0. tree of G Clearly, for every edge in T we can assume that the two corresponding ˜ split(W ) are in the same side of the bipartition. If for every edge loops in G f in W , the two corresponding loops in F˜ split(W ) are in the same side of the bipartition, then we are done by assigning to f to that side of the bipartition. Let W + ⊆ W contain the edges such that the two corresponding loops in ˜ split(W ) are in the same side of the bipartition, and let W − = W \ W + . G Thus E(T ) ⊆ W + . Thus we only need to show that W = W + . By contradiction, among all the edges in W − , let vw be one that mini˜ split(W ) by mizes the distance between v and w in the graph obtained from G + replacing each pair of loops corresponding to an edge in W with the original ˜ Notice that for every edge in W − , such distance is always finite, edge in G. since T ⊆ W + . Moreover let P be the path from v to w that realizes this ˜ corresponding minimum length. Now let C = v, vw, P, v be the cycle in G to the edges in P and the edge vw. By the minimality of vw it follows that the cycle C has no chords in F˜ . ˜ split(W ) to a family S of paths Notice that the cycle C corresponds in G split(W ) whose first and last edge are loops in F˜ corresponding to edges in ˜ split(W ) are not in the W . Since the two loops corresponding to vw in G same side of the bipartition, it follows that in S there is an odd number of paths whose first and last edge are in different sides of the bipartition. By ClaimP4.23, this means that in S there is an odd number of paths S such that vw∈E(S) (σv,e + σw,e ) ≡2 2. But this implies that C is an odd cycle of ˜ that contains edges in F˜ , a contradiction. ¦ G

Bibliography [1] P. Camion, Caract´erisation des matrices unimodulaires, Cahiers du Cen´ tre d’Etudes de Recherche Op´erationnelle 5 (1963) 181-190. [2] P. Camion, Matrices totalement unimodulaires et probl`emes combinaioires, Ph.D. Thesis, Universit´e Libre de Bruxelles, Brussels, 1963. [3] P. Camion, Characterizations of totally unimodular matrices, Proceedings of the American Mathematical Society 16 (1965) 1068-1073. [4] R. Chandrasekaran, Polynomial algorithms for totally dual integral systems and extensions, in Studies on Graphs and Discrete Programming (P. Hansen, ed.), Annals of Discrete Mathematics 11 (1981) 39-51. [5] V. Chvatal, Edmonds polytopes and a hierarchy of combinatorial problems, Discrete Mathematics 4 (1973) 305-337. [6] M. Conforti, M. Di Summa, F. Eisenbrand, and L.A. Wolsey, Network formulations of mixed-integer programs, accepted in Mathematics of Operations Research, 2006. [7] M. Conforti, A.M.H. Gerards, and G. Zambelli, Mixed-integer vertex covers on bipartite graphs, in Integer Programming and Combinatorial Optimization (M. Fischetti and D.P. Williamson eds.), proceedings of IPCO 2007, LNCS Vol. 4513, Springer, 2007, 324-336. ´ Tardos, Sensitivity the[8] W. Cook, A.M.H. Gerards, A. Schrijver, and E. orems in integer linear programming, Mathematical Programming 34 (1986) 251-264. [9] G. Cornu´ejols, J. Fonlupt, and D. Naddef, The Traveling Salesman Problem on a Graph and some Related Integer Polyhdra, Mathematical Programming 33 (1985) 1–27. [10] A. Del Pia, A. Musitelli, and G. Zambelli, A class of matrices with the Edmonds-Johnson property, manuscript, 2008.

100

BIBLIOGRAPHY

[11] A. Del Pia and G. Zambelli, A class of matrices arising from bidirected graphs: total dual integrality and strong Chv´atal rank, submitted, 2007. [12] A. Del Pia and G. Zambelli, A class of matrices with Strong Chv´atal Rank 1, submitted, 2008. [13] J. Edmonds, Maximum matching and a polyhedron with 0, 1-vertices, Journal of Research of the National Bureau of Standards (B) 69 (1965) 125-130. [14] J. Edmonds and R. Giles, A min-max relation for submodular functions on graphs, Annals of Discrete Mathematics 1 (1977) 185-204. [15] J. Edmonds and E.L. Johnson, Matching: a well-solved class of integer linear programs, in Combinatorial Structures and Their Applications (R.K. Guy, et al., eds.), Gordon and Breach, New York, 1970, 89-92. [16] J. Edmonds and E.L. Johnson, Matching, Euler tours and the Chinese postman, Mathematical Programming 5 (1973) 88-124. [17] F. Eisenbrand, On the membership problem for the elementary closure of a polyhedron, Combinatorica 19 (1999) 297-300. [18] A.M.H. Gerards, personal communication, 2007. [19] A.M.H. Gerards and A. Schrijver, Matrices with the Edmonds-Johnson property, Combinatorica 6 (1986) 365-379. [20] A. Ghouila-Houri, Charact´erisations des Matrices Totalement Unimodulaires, Comptes Rendus de l’Acad´emie des Sciences 254 (1962) 11921193. [21] F.R. Giles, and W.R. Pulleyblank, Total dual integrality and integer polyhedra, Linear Algebra and Its Applications 25 (1979) 191-196. [22] M.X. Goemans, Minimum Bounded Degree Spanning Trees, Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (2006) 273-282. [23] M. Gr¨otschel, L. Lov´asz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin, 1988. [24] I. Heller and C.B. Tompkins, An extension of a theorem of Danzig’s, in Linear Inequalities and Related Systems (H. W. Kuhn and A. W. Tucker, eds.) Princeton University Press, Princeton, N.J., 1956, 247-254.

BIBLIOGRAPHY

101

[25] A.J. Hoffman and J.B. Kruskal, Integral boundary points of convex polyhedra, in Linear Inequalities and Related Systems (H. W. Kuhnand A. W. Tucker, eds.), Princeton Univ. Press, Princeton, N.J., 1956, 223-246. ´ Tardos, How to Tidy Up [26] C.A.J. Hurkens, L. Lov´asz, A. Schrijver, and E. your Set System?, in: Combinatorics, Colloquia Mathematica Societatis J´anos Bolyai, 52, North-Holland (1998) 309-314. [27] K. Jain, A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem, Combinatorica 21 (2001) 39-60. [28] R.R. Meyer, On the existence of optimal solutions to integer and mixedinteger programming problems, Mathematical Programming 7 (1974) 223-235. [29] A. Schrijver, Theory of Linear and Integer Programming, Wiley, New York, 1986. [30] A. Schrijver, Combinatorial Optimization. Polyhedra and Efficiency, Springer-Verlag, Berlin-Heidelberg, 2003. [31] M. Singh, Iterative methods in Combinatorial optimization, Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, 2008. ´ Tardos, A strongly polynomial minimum cost circulation algorithm, [32] E. Combinatorica 5 (1985) 247-255. ´ Tardos, A strongly polynomial algorithm to solve combinatorial linear [33] E. programs, Operations Research 34 (1986) 250-256. [34] D.B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall, Upper Saddle River NJ, 2001.